Class 12 Mathematics Chapter 3 Binary Operations

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Class 12 Mathematics Chapter 3 Binary Operations

This quiz on Chapter 3: Binary Operations for ICSE Class 12 Mathematics is designed to test students’ understanding of fundamental concepts and properties of binary operations. It includes questions that assess conceptual clarity, logical reasoning, and application skills related to operations defined on various sets such as real numbers, integers, and matrices. Students will encounter problems involving commutativity, associativity, identity, and inverse elements, along with determining whether a given operation is closed on a set. The quiz also features analytical questions where learners must verify properties and construct examples of binary operations satisfying specific conditions. By attempting this quiz, students can strengthen their grasp of abstract algebraic structures and enhance their problem-solving proficiency in preparation for board examinations.

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(A) For the set $\{0, 1, 2, 3, 4, 5\}$ under the binary operation $*$ defined as
$$
a * b =
\begin{cases}
a + b & \text{if } a + b < 6a + b - 6 & \text{if } a + b \geq 6 \end{cases}, $$ the inverse of $4$ is $2$. (R) The inverse of an element $x$ in this operation must satisfy $x * x^{-1} = x^{-1} * x = 0$, where $0$ is the identity element.

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(A) For the set $A = \{0, 1, 2, 3, 4, 5\}$ with the operation $*$ defined by $a * b = \begin{cases} a + b & \text{if } a + b < 6, \\ a + b - 6 & \text{if } a + b \geq 6, \end{cases}$, the element $0$ is the identity because its row and column replicate the topmost row and leftmost column of the composition table. (R) The first row and first column of the composition table must match the topmost row and leftmost column for an element to be the identity.

3 / 300

(A) A binary operation $*$ defined on a set $A = \{1, 2, 3, 4\}$ with the composition table symmetric about the main diagonal implies that $(2 * 4) = (4 * 2)$.
(R) The symmetry of the composition table about the main diagonal is both necessary and sufficient for a binary operation to be commutative.

4 / 300

(A) For the set $S = \{1, -1, i, -i\}$ under multiplication, the operation table shows that all entries are elements of $S$.
(R) The closure property requires that for any binary operation * on a set $S$, the result of $a * b$ must belong to $S$ for every $a, b \in S$.

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(A) The binary operation $*$ defined on the set $S = \{a, b, c\}$ by the following table is commutative.

(R) The composition table for $*$ is symmetric about the principal diagonal.

$*$ | $a$ | $b$ | $c$
--- | --- | --- | ---
$a$ | $a$ | $b$ | $c$
$b$ | $b$ | $a$ | $b$
$c$ | $c$ | $b$ | $a$

6 / 300

(A) In the set $S = \{0, 1, 2, 3, 4, 5\}$ with the binary operation $*$ defined as $a * b = a + b$ if $a + b < 6$, and $a * b = a + b - 6$ if $a + b \geq 6$, the element $3$ has an inverse. (R) For the given binary operation, every non-zero element $a$ in $S$ has an inverse of the form $6 - a$.

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(A) In a composition table for a binary operation $*$ on set $A = \{1, -1\}$, if the row and column corresponding to an element $e \in A$ replicate the headers, then $e$ must be the identity element.

(R) The identity element $e$ satisfies $a * e = e * a = a$ for all $a \in A$, which is reflected in the replication of headers in its row and column in the composition table.

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(A) A binary operation $*$ on a set $S$ is commutative if and only if for all $a, b \in S$, $a*b = b*a$.
(R) The composition table of a commutative binary operation is symmetric about the main diagonal.

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(A) The set $S = \{1, -1, i, -i\}$ under multiplication satisfies the closure property because all entries in the composition table belong to $S$.
(R) For any two elements $a, b \in S$, the product $a * b$ must be an element of $S$ for the closure property to hold.

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(A) For a set $A = \{1, -1\}$ with the operation * defined by multiplication, the element 1 is the identity element.
(R) In the composition table for the operation *, the row and column headed by 1 match the topmost row and leftmost column respectively.

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(A) For the set $\mathbb{Q}$ of rational numbers under addition, every element has an additive inverse.
(R) The identity element for addition on $\mathbb{Q}$ is 0, and for any rational number $a$, there exists $-a$ such that $a + (-a) = (-a) + a = 0$.

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(A) In the composition table for the set $A = \{-1, 1\}$ under multiplication, the identity element is $1$.
(R) The row and column headings of the table replicate when the identity element is used as a heading.

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(A) If a binary operation $*$ on a set $S$ has a composition table that is symmetric about the main diagonal, then $*$ is commutative.
(R) A binary operation is commutative if and only if its composition table is symmetric about the main diagonal.

14 / 300

(A) The operation $*$ defined on the set $S = \{1, \omega, \omega^2\}$ by multiplication is closed because all entries in the multiplication table belong to $S$.
(R) For any two elements $a, b \in S$, the product $a * b$ must be an element of $S$ for the operation to satisfy closure.

15 / 300

(A) The operation $*$ defined on the set $A = \{1, 2, 3\}$ by $a * b = \text{min}\{a, b\}$ is a binary operation.

(R) All entries in the composition table for $*$ are elements of the set $A$.

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(A) The binary operation $*$ defined on the set $S = \{a, b, c\}$ by the following composition table is commutative and has an identity element:

$\begin{array}{c|ccc} \\ * & a & b & c \\ \\ \hline \\ a & a & b & c \\ \\ b & b & a & c \\ \\ c & c & c & c \\ \\ \end{array}$

(R) A binary operation is commutative if its composition table is symmetric about the principal diagonal, and it has an identity element if there exists an element $e \in S$ such that $a * e = e * a = a$ for all $a \in S$.

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(A) The binary operation $*$ defined on the set $S = \{a, b, c\}$ by the given composition table is commutative.
(R) The composition table for $*$ is symmetric about the principal diagonal.

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(A) A binary operation defined by a composition table on a finite set is always commutative if the table is symmetric about the principal diagonal.
(R) A composition table that is symmetric about the principal diagonal ensures that $a*b = b*a$ for all elements $a, b$ in the set.

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(A) For the binary operation $*$ defined on the set of real numbers $\mathbb{R}$ as $a * b = a + b - ab$, if an element $a \in \mathbb{R}$ has an inverse, then it must satisfy $a \neq 1$.
(R) The identity element for the binary operation $*$ is $0$, and the inverse condition requires solving $a * b = 0$ which simplifies to $b = \frac{a}{a - 1}$ for $a \neq 1$.

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(A) For every element $a$ in a set $S$ with respect to a binary operation $*$, if an inverse exists, it must be unique.
(R) The uniqueness of the inverse element is guaranteed because assuming two distinct inverses leads to a contradiction.

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(A) The number $-3$ has an additive inverse in the set of integers.
(R) For any integer $a$, its additive inverse is $-a$ such that $a + (-a) = 0$.

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(A) For the operation $*$ defined on $\mathbb{R}$ by $a * b = a + b - ab$, the identity element exists and is equal to 0.
(R) The identity element for any binary operation must satisfy $e * a = a * e = a$ for all $a$ in the set.

23 / 300

(A) The operation * defined on the set of rational numbers excluding 1 by $a * b = a + b - ab$ has an identity element.
(R) For the given operation, there exists an element $e \in \mathbb{Q} - \{1\}$ such that $a * e = e * a = a$ for all $a \in \mathbb{Q} - \{1\}$.

24 / 300

(A) For the operation $*$ defined on $\mathbb{R}$ by $a * b = a + b - ab$, the identity element is 1.
(R) The identity element $e$ satisfies $a * e = e * a = a$ for every $a \in S$.

25 / 300

(A) The operation $\circ$ defined by $a \circ b = a + 2b$ on the set of real numbers is distributive over the operation $*$ defined by $a * b = |a - b|$.

(R) For any operations $\circ$ and $*$ on a set, if $\circ$ is distributive over $*$, then $a \circ (b * c) = (a \circ b) * (a \circ c)$ must hold for all $a, b, c$.

26 / 300

(A) The binary operation $*$ defined by $a * b = a^2 + b^2$ for all $a, b \in \mathbb{R}$ is associative.
(R) For any real numbers $a, b, c$, the equality $(a * b) * c = a * (b * c)$ holds under the operation $*$.

27 / 300

(A) For any sets $A, B, C$, the operation of intersection is both associative and distributive over union.
(R) The operations of union and intersection of sets satisfy the distributive property $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ for all sets $A, B, C$.

28 / 300

(A) The binary operation $*$ defined on the set of real numbers $\mathbb{R}$ by $a * b = a + b - ab$ is commutative.
(R) For all $a, b \in \mathbb{R}$, $a + b - ab = b + a - ba$.

29 / 300

(A) The binary operation $*$ defined on $\mathbb{R}$ by $a * b = a^2 + b^2$ is commutative.
(R) For all $a, b \in \mathbb{R}$, $a^2 + b^2 = b^2 + a^2$.

30 / 300

(A) The binary operation $*$ defined by $a * b = a + 2b$ is distributive over addition on $\mathbb{R}$.
(R) For all $a, b, c \in \mathbb{R}$, the condition $a * (b + c) = (a * b) + (a * c)$ holds true.

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(A) The binary operation defined by $x * y = x + 2y$ on the set of real numbers is not associative.
(R) For this operation, $(1 * 2) * 3 \neq 1 * (2 * 3)$.

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(A) The operation of union of sets is both associative and distributive over intersection.
(R) For any three sets $A$, $B$, and $C$, the following holds:
$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
and
$(A \cup B) \cup C = A \cup (B \cup C).$

33 / 300

(A) The binary operation $*$ defined on $\mathbb{Q}$ by $a * b = \frac{a + b}{2}$ is commutative.
(R) For any $a, b \in \mathbb{Q}$, $\frac{a + b}{2} = \frac{b + a}{2}$ due to the commutative property of addition.

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(A) The binary operation $*$ defined on $\mathbb{R}$ by $a * b = a + 2b$ is associative.
(R) A binary operation that is commutative must also be associative.

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(A) Multiplication distributes over addition in real numbers.
(R) For all $a, b, c \in \mathbb{R}$, the equation $a(b + c) = a \cdot b + a \cdot c$ holds true.

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(A) The binary operation $*$ defined by $a * b = a + b + ab$ on the set of real numbers is associative.
(R) For any real numbers $a, b, c$, $(a * b) * c = a * (b * c)$ holds true.

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(A) The union of sets $A$ and $B$ is equal to the union of sets $B$ and $A$.
(R) Union of sets is a commutative binary operation.

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(A) Addition of real numbers is a commutative operation.
(R) For all $a, b \in \mathbb{R}$, we have $a + b = b + a$.

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(A) The binary operation $*$ defined on $\mathbb{R}$ by $a * b = a + b$ is commutative.
(R) For any two real numbers $a$ and $b$, $a + b = b + a$.

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(A) The binary operation * defined on $\mathbb{R} - \{-1\}$ by $a * b = \frac{a + b}{b + 1}$ is associative.
(R) For any $a, b, c \in \mathbb{R} - \{-1\}$, $(a * b) * c = a * (b * c)$.

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(A) The binary operation * defined on the set of real numbers by $a * b = a + b - ab$ is commutative.
(R) For any two real numbers $a$ and $b$, $a + b - ab = b + a - ba$.

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(A) Addition on the set of integers $\mathbb{Z}$ is commutative.
(R) For all $a, b \in \mathbb{Z}$, $a + b = b + a$.

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(A) The number of binary operations on a set with 3 elements is 512.
(R) For a finite set with $n$ elements, the number of possible binary operations is $n^{n^2}$.

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(A) The number of binary operations on a set with 3 elements is 512.
(R) For a finite set $A$ with $n$ elements, the number of binary operations is $n^{n^2}$.

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(A) The number of binary operations on a set with 3 elements is 27.
(R) For a set with $n$ elements, the number of binary operations is $n^{n^2}$.

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(A) The number of binary operations on a set with 3 elements is $3^9$.
(R) For a finite set $A$ with $n$ elements, the number of possible binary operations on $A$ is $n^{n^2}$.

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(A) The set $S = \{x \in \mathbb{Z}^+ \mid x \leq 5\}$ under the binary operation $*$ defined by $a * b = \text{max}(a, b)$ is closed.
(R) For any two elements $a, b \in S$, the result of $a * b$ always belongs to $S$.

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(A) The number of possible binary operations on a set with 3 elements is 19683.
(R) For a set with $n$ elements, the total number of binary operations is given by $n^{n^2}$.

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(A) The set $A = \{[(x, y), p] \mid x \in Z, y \in Z, p \in Z^* \text{ and } x + y = p\}$ represents a binary operation on the set of integers.
(R) A binary operation can always be represented as a set of ordered pairs where the first element is an input pair and the second element is the result of the operation.

50 / 300

(A) The binary operation $*$ defined on $\mathbb{Q}$ by $a * b = ab + 1$ is associative.
(R) For all $a, b, c \in \mathbb{Q}$, $(a * b) * c = a * (b * c)$.

51 / 300

(A) The number of binary operations on a set with 4 elements is $4^{16}$.
(R) For any finite set with $n$ elements, the number of binary operations is given by $n^{n^2}$.

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(A) The operation $a * b = \max\{a, b\}$ is a binary operation on the set of real numbers $\mathbb{R}$.
(R) For any two real numbers $a$ and $b$, $\max\{a, b\}$ is always a real number and belongs to $\mathbb{R}$.

53 / 300

(A) The number of binary operations on a set with 4 elements is $4^{16}$.
(R) A binary operation on a set $A$ with $n$ elements is a function from $A \times A$ to $A$, and the number of such functions is $n^{(n^2)}$.

54 / 300

(A) The set $A = \{[(x, y), p] \mid x, y \in \mathbb{Z}, p \in \mathbb{Z}^+ \text{ and } x + y = p\}$ represents a binary operation on the set of integers $\mathbb{Z}$.

(R) A binary operation on a set $S$ is a function $f: S \times S \rightarrow S$ that is closed under the operation.

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(A) The binary operation $*$ defined by $a * b = a^b$ on the set of positive integers is commutative.
(R) For any two positive integers $a$ and $b$, $a^b = b^a$.

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(A) The number of binary operations on a set with 4 elements is $4^{16}$.

(R) For a finite set $A$ with $n$ elements, the number of binary operations is given by $n^{(n^2)}$.

57 / 300

(A) Addition is a binary operation on the set of positive integers \$(Z^*)\$.
(R) A binary operation maps a pair of elements from a set to a single element within the same set.

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(A) The number of binary operations on a set with 3 elements is 512.
(R) For a set A with n elements, the total number of binary operations is $n^{n^2}$.

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(A) A binary operation on a set $S$ can be represented as a function from $S \times S$ to $S$.

(R) The closure property ensures that the output of the binary operation lies within the set $S$.

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(A) Addition of real numbers is a binary operation because it takes two real numbers and returns another real number.
(R) A binary operation on a set $S$ is defined as a function from $S \times S$ to $S$.

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(A) The binary operation * defined by $a * b = \text{LCM}\{a, b\}$ on the set $A = \{1, 2, 3, 4, 5\}$ has an identity element.
(R) For any finite set $A$ under the LCM operation, the element $1$ acts as the identity because $\text{LCM}\{a, 1\} = a$ for all $a \in A$.

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(A) Subtraction is a binary operation on the set of natural numbers ℕ.
(R) For all $a, b \in ℕ$, $a - b$ always belongs to ℕ.

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(A) The operation $+$ is a binary operation on the set of odd integers.
(R) The sum of two odd integers is always an odd integer.

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(A) The operation $a * b = \frac{a + b}{2}$ defined on the set of even integers is a binary operation.
(R) The sum and average of any two even integers is always an even integer.

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(A) If $f: A \rightarrow B$ and $g: B \rightarrow C$ are bijective functions, then $(g \circ f)^{-1}$ exists and is equal to $g^{-1} \circ f^{-1}$.
(R) The composition of two bijective functions is always bijective.

66 / 300

(A) The operation * defined by $a * b = \text{LCM}\{a, b\}$ is a binary operation on the set $A = \{1, 2, 3, 4, 5\}$.
(R) For any two elements $a, b \in A$, the LCM of $a$ and $b$ always belongs to $A$.

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(A) Subtraction is a binary operation on $\mathbb{N}$.
(R) For any $a, b \in \mathbb{N}$, $(a - b) \in \mathbb{N}$.

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(A) The sum of any two odd natural numbers is an even natural number.
(R) The set of odd natural numbers is not closed under addition.

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(A) Addition is a binary operation on the set of even natural numbers.
(R) The sum of two even natural numbers is always an even natural number.

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(A) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by $f(x) = 2x + 3$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ is defined by $g(x) = \frac{x - 3}{2}$, then $(g \circ f)(x) = x$ for all $x \in \mathbb{R}$.
(R) The function $g$ is the inverse of $f$.

71 / 300

(A) The binary operation * on set $A = \{1, 2, 3\}$ defined by $a * b = \text{LCM} \{a, b\}$ is commutative.
(R) $\text{LCM} \{a, b\} = \text{LCM} \{b, a\}$ for all $a, b \in A$.

72 / 300

(A) Subtraction is a binary operation on ℕ.
(R) For some $a, b \in ℕ$, $a - b$ may not belong to ℕ.

73 / 300

(A) Addition is a binary operation on the set of odd natural numbers.
(R) The sum of any two odd natural numbers is an even natural number.

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(A) Addition is a binary operation on the set of even natural numbers.
(R) The sum of two even natural numbers is always an even natural number.

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(A) The composition $(g \circ f)(x)$ is defined as $g(f(x))$ for functions $f: A \rightarrow B$ and $g: B \rightarrow C$.
(R) The domain of $g$ must match the codomain of $f$ for the composition $g \circ f$ to be defined.

76 / 300

Let $*$ be a binary operation on the set $S = \{a, b, c, d\}$ with the following composition table:
$\begin{array}{c|cccc} \\ * & a & b & c & d \\ \\ \hline \\ a & a & b & c & d \\ \\ b & b & a & d & c \\ \\ c & c & d & b & a \\ \\ d & d & c & a & b \\ \\ \end{array}$
Which element is the inverse of $c$?

77 / 300

Consider the binary operation $*$ on the set $S = \{0, 1, 2, 3, 4\}$ defined as:
$a * b = \\ \begin{cases} \\ a + b & \text{if } a + b < 5 \\ \\ a + b - 5 & \text{if } a + b \geq 5 \\ \end{cases}$ What is the inverse of the element $3$ under this operation?

78 / 300

Let $*$ be a binary operation on the set $S = \{1, 2, 3, 4, 5\}$ defined by $a * b = \text{max}(a, b)$. Which of the following elements has an inverse in $S$?

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Given the set $A = \{a, b, c, d\}$ with the binary operation $\circ$ defined by the composition table below, which elements are invertible? Assume the identity element is $a$.

$\begin{array}{c|cccc} \\ \circ & a & b & c & d \\ \\ \hline \\ a & a & b & c & d \\ \\ b & b & a & d & c \\ \\ c & c & d & a & b \\ \\ d & d & c & b & a \\ \\ \end{array}$

80 / 300

Using the same composition table as above ($S = \{0, 1, 2, 3\}$ with $\ast$), which element is the inverse of $2$ under the operation $\ast$ if the identity element is $0$?

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Consider the set $S = \{0, 1, 2, 3\}$ with the binary operation $\ast$ defined by the following composition table:

$\begin{array}{c|cccc} \\ \ast & 0 & 1 & 2 & 3 \\ \\ \hline \\ 0 & 0 & 1 & 2 & 3 \\ \\ 1 & 1 & 0 & 3 & 2 \\ \\ 2 & 2 & 3 & 0 & 1 \\ \\ 3 & 3 & 2 & 1 & 0 \\ \\ \end{array}$

What is the identity element of this set under the operation $\ast$?

82 / 300

Let * be an operation defined on the set $C = \{1, 2, 3, 4\}$ with the following table:

$*$ | $1$ | $2$ | $3$ | $4$
$1$ | $1$ | $2$ | $3$ | $4$
$2$ | $2$ | $4$ | $1$ | $3$
$3$ | $3$ | $1$ | $4$ | $2$
$4$ | $4$ | $3$ | $2$ | $1$

Which of the following statements is correct?

83 / 300

Given the operation table for * on the set $B = \{p, q, r\}$:

$*$ | $p$ | $q$ | $r$
$p$ | $p$ | $q$ | $r$
$q$ | $r$ | $p$ | $q$
$r$ | $q$ | $r$ | $p$

Which statement is true about this operation?

84 / 300

Consider the operation table for * on the set $A = \{x, y, z\}$:

$*$ | $x$ | $y$ | $z$
$x$ | $x$ | $y$ | $z$
$y$ | $y$ | $z$ | $x$
$z$ | $z$ | $x$ | $y$

Which of the following properties does the operation * satisfy?

85 / 300

Given $U = \{a, b\}$ with two operations $\oplus$ and $\otimes$ defined below:

\begin{tabular}{|c|cc|}
\hline
$\oplus$ & a & b

\hline
a & a & b

b & b & a

\hline
\end{tabular}
\quad
\begin{tabular}{|c|cc|}
\hline
$\otimes$ & a & b

\hline
a & a & a

b & a & c

\hline
\end{tabular}

Which statement correctly compares the closure properties?

86 / 300

Consider the set $T = \{x, y, z\}$ with binary operation $\circ$ given by the composition table:

\begin{tabular}{|c|ccc|}
\hline
$\circ$ & x & y & z

\hline
x & x & z & y

y & z & y & x

z & y & x & w

\hline
\end{tabular}

What can be concluded about the closure property of $\circ$ on $T$?

87 / 300

Let $S = \{0, 1, 2\}$ and $\ast$ be a binary operation defined by the following composition table:

\begin{tabular}{|c|ccc|}
\hline
$\ast$ & 0 & 1 & 2

\hline
0 & 0 & 1 & 2

1 & 1 & 2 & 0

2 & 2 & 0 & 1

\hline
\end{tabular}

Which of the following statements is true regarding the closure property of $\ast$ on $S$?

88 / 300

Given the operation $*$ on the set $S = \{p, q, r\}$ with the following table:
\begin{center}
\begin{tabular}{|c|ccc|}
\hline
$*$ & p & q & r

\hline
p & p & q & r

q & r & p & q

r & q & r & p

\hline
\end{tabular}
\end{center}
Which of the following correctly evaluates $(p * q) * r$?

89 / 300

Let $*$ be a binary operation on the set $A = \{a, b, c, d\}$ defined by the following table:
\begin{center}
\begin{tabular}{|c|cccc|}
\hline
$*$ & a & b & c & d

\hline
a & a & b & c & d

b & b & a & d & c

c & c & d & a & b

d & d & c & b & a

\hline
\end{tabular}
\end{center}
What is the inverse of the element $c$?

90 / 300

Consider the operation $*$ defined on the set $A = \{1, 2, 3, 4\}$ with the following composition table:
\begin{center}
\begin{tabular}{|c|cccc|}
\hline
$*$ & 1 & 2 & 3 & 4

\hline
1 & 1 & 2 & 3 & 4

2 & 2 & 4 & 1 & 3

3 & 3 & 1 & 4 & 2

4 & 4 & 3 & 2 & 1

\hline
\end{tabular}
\end{center}
Which of the following statements is true about this operation?

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On the set $A = \{-2, -1, 0, 1, 2\}$, a binary operation $*$ is defined as $a * b = \max\{a, b\}$. Which elements are invertible?

92 / 300

Consider the binary operation $*$ on the set $\mathbb{Q} - \{1\}$ defined by:
$a * b = a + b - ab$
What is the inverse of 2 under this operation?

93 / 300

Let $*$ be a binary operation on the set $S = \{0, 1, 2, 3, 4\}$ defined by:
$a * b = \begin{cases} \\ a + b & \text{if } a + b < 5 \\ \\ a + b - 5 & \text{if } a + b \geq 5 \\ \end{cases}$ Which of the following is the inverse of 3 with respect to $*$?

94 / 300

Given the composition table for a binary operation $\circ$ on the set $S = \{p, q, r\}$:
$\begin{array}{c|ccc} \\ \circ & p & q & r \\ \hline \\ p & p & q & r \\ \\ q & q & r & p \\ \\ r & r & p & q \\ \\ \end{array}$
Is the operation $\circ$ commutative?

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For the set $A = \{1, 2, 3, 4\}$ under multiplication modulo 5, which element(s) are invertible? Use the following partial composition table:
$\begin{array}{c|cccc} \\ \times_5 & 1 & 2 & 3 & 4 \\ \hline \\ 1 & 1 & 2 & 3 & 4 \\ \\ 2 & 2 & 4 & 1 & 3 \\ \\ 3 & 3 & 1 & 4 & 2 \\ \\ 4 & 4 & 3 & 2 & 1 \\ \\ \end{array}$

96 / 300

Consider the composition table for a binary operation $*$ on the set $A = \{a, b, c\}$:
$\begin{array}{c|ccc} \\ * & a & b & c \\ \hline \\ a & a & b & c \\ \\ b & b & b & c \\ \\ c & c & c & c \\ \\ \end{array}$
What is the identity element of $(A, *)$?

97 / 300

Consider the binary operation defined by the following table on the set $\{x, y, z\}$. Which pair confirms that the operation is NOT commutative?

$*$ | $x$ | $y$ | $z$
--- | --- | --- | ---
$x$ | $x$ | $y$ | $z$
$y$ | $x$ | $y$ | $z$
$z$ | $y$ | $y$ | $z$

98 / 300

For the binary operation table defined on the set $\{p, q, r\}$ below, which of the following statements correctly describes its commutativity?

$*$ | $p$ | $q$ | $r$
--- | --- | --- | ---
$p$ | $p$ | $q$ | $r$
$q$ | $r$ | $q$ | $p$
$r$ | $q$ | $r$ | $p$

99 / 300

Given the binary operation table below for a set $S = \{a, b, c\}$, determine if the operation is commutative.

$*$ | $a$ | $b$ | $c$
--- | --- | --- | ---
$a$ | $a$ | $b$ | $c$
$b$ | $b$ | $c$ | $a$
$c$ | $c$ | $a$ | $b$

100 / 300

Given the set $S = \{e, a, b, c\}$ with identity element $e$ and the operation * defined by the table:
\begin{tabular}{|c|c|c|c|c|}
\hline
* & e & a & b & c
\hline
e & e & a & b & c
\hline
a & a & e & c & b
\hline
b & b & c & e & a
\hline
c & c & b & a & e
\hline
\end{tabular}
Which element is the inverse of $b$?

101 / 300

Let $S = \{a, b, c\}$ and the binary operation * be defined by the following composition table:
\begin{tabular}{|c|c|c|c|}
\hline
* & a & b & c
\hline
a & a & b & c
\hline
b & b & a & b
\hline
c & c & b & a
\hline
\end{tabular}
Is the operation * commutative on $S$?

102 / 300

Consider the set $S = \{0, 1, 2\}$ under the binary operation * defined by the composition table below:
\begin{tabular}{|c|c|c|c|}
\hline
* & 0 & 1 & 2
\hline
0 & 0 & 0 & 0
\hline
1 & 0 & 1 & 2
\hline
2 & 0 & 2 & 1
\hline
\end{tabular}
Does the set $S$ satisfy the closure property under *?

103 / 300

Let $*$ be a binary operation on the set $B = \{1, 2, 3\}$ given by the following table:
$\begin{array}{c|ccc} \\ * & 1 & 2 & 3 \\ \\ \hline \\ 1 & 1 & 2 & 3 \\ \\ 2 & 2 & 1 & 3 \\ \\ 3 & 3 & 3 & 1 \\ \end{array}$
Which of the following elements has an inverse in $B$ under $*$?

104 / 300

Given the binary operation $*$ on the set $A = \{x, y, z\}$ defined by the following table:
$\begin{array}{c|ccc} \\ * & x & y & z \\ \\ \hline \\ x & x & y & z \\ \\ y & y & y & z \\ \\ z & z & z & x \\ \end{array}$
What is the identity element of $A$ under $*$?

105 / 300

Consider the following composition table for a binary operation $*$ on the set $S = \{p, q, r\}$.
$\begin{array}{c|ccc} \\ * & p & q & r \\ \\ \hline \\ p & p & q & r \\ \\ q & q & q & r \\ \\ r & r & r & p \\ \end{array}$
Is the operation $*$ commutative?

106 / 300

Which of the following rational numbers has no multiplicative inverse in the set of natural numbers $\mathbb{N}$?

107 / 300

What is the additive inverse of the real number $5$ under the standard addition operation?

108 / 300

Let $*$ be a binary operation on a non-empty set $S$ with identity element $e$. What condition must an element $b \in S$ satisfy to be the inverse of $a \in S$?

109 / 300

For the set $A = \{1, 2, 3\}$ and operation $a * b = \text{LCM}(a, b)$, does $*$ form a valid binary operation?

110 / 300

In the set $A = \{0, 1, 2\}$ under addition modulo $3$, which elements are invertible?

111 / 300

For a set $A = \{1, -1\}$ under multiplication, which element is the identity element?

112 / 300

For $S = \{p, q\}$, if $p*q = q*p$ in a binary operation table, what does this imply?

113 / 300

Which of these composition tables represents a commutative operation?

114 / 300

Given the following binary operation table for set $S = \{x, y\}$, is the operation commutative?

$*$ | $x$ | $y$
--- | --- | ---
$x$ | $x$ | $y$
$y$ | $y$ | $x$

115 / 300

What is the identity element for the set $S = \{1, -1, i, -i\}$ under multiplication?

116 / 300

For the set $S = \{1, -1, i, -i\}$, which of the following demonstrates the commutative property of multiplication?

117 / 300

Consider the set $S = \{1, -1, i, -i\}$ with multiplication as the binary operation. Which of the following products is NOT in $S$?

118 / 300

Given the operation table on set $B = \{m, n, o\}$:

$*$ | $m$ | $n$ | $o$
--- | --- | --- | ---
$m$ | $m$ | $n$ | $o$
$n$ | $n$ | $o$ | $m$
$o$ | $o$ | $m$ | $n$

What is the identity element for the operation $*$?

119 / 300

Consider the binary operation table on set $A = \{p, q, r\}$:

$*$ | $p$ | $q$ | $r$
--- | --- | --- | ---
$p$ | $p$ | $q$ | $r$
$q$ | $q$ | $r$ | $p$
$r$ | $r$ | $p$ | $q$

Is the operation $*$ commutative?

120 / 300

Given the following binary operation table on set $S = \{x, y, z\}$:

$*$ | $x$ | $y$ | $z$
--- | --- | --- | ---
$x$ | $x$ | $y$ | $z$
$y$ | $y$ | $x$ | $y$
$z$ | $z$ | $y$ | $x$

What is the value of $y * z$?

121 / 300

Consider the binary operation $*$ on the set $\{2, 4, 6, 8\}$ defined by $a * b = \text{HCF}(a, b)$. What is the value of $(2 * 4) * (6 * 8)$?

122 / 300

Let $*$ be a binary operation on the set $A = \{x, y, z\}$ defined by the following table:

| $*$ | x | y | z |
|-----|---|---|---|
| x | y | x | z |
| y | x | y | z |
| z | z | z | z |

Which of the following statements is true?

123 / 300

Consider a binary operation $*$ defined on the set $S = \{1, 2, 3, 4\}$ by the following composition table:

| $*$ | 1 | 2 | 3 | 4 |
|-----|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 1 | 3 |
| 3 | 3 | 1 | 4 | 2 |
| 4 | 4 | 3 | 2 | 1 |

Which of the following properties does this operation satisfy?

124 / 300

Given the binary operation $*$ on the set $Q = \{p, q, r\}$ with the following table, is the operation associative?

$\begin{array}{|c|c|c|c|} \\ \hline \\ * & p & q & r \\ \\ \hline \\ p & p & p & p \\ \\ \hline \\ q & p & q & r \\ \\ \hline \\ r & p & r & q \\ \\ \hline \\ \end{array}$

125 / 300

Consider the following composition table for a binary operation $*$ on the set $P = \{1, 2, 3, 4\}$. Which element serves as the identity?

$\begin{array}{|c|c|c|c|c|} \\ \hline \\ * & 1 & 2 & 3 & 4 \\ \\ \hline \\ 1 & 1 & 2 & 3 & 4 \\ \\ \hline \\ 2 & 2 & 1 & 4 & 3 \\ \\ \hline \\ 3 & 3 & 4 & 1 & 2 \\ \\ \hline \\ 4 & 4 & 3 & 2 & 1 \\ \\ \hline \\ \end{array}$

126 / 300

Below is the composition table for a binary operation $*$ on the set $S = \{a, b, c\}$. Is the operation commutative?

$\begin{array}{|c|c|c|c|} \\ \hline \\ * & a & b & c \\ \\ \hline \\ a & a & b & c \\ \\ \hline \\ b & b & a & c \\ \\ \hline \\ c & c & c & b \\ \\ \hline \\ \end{array}$

127 / 300

Consider the binary operation table on $B = \{x, y\}$:
$\begin{array}{c|cc} \\ * & x & y \\ \\ \hline \\ x & x & y \\ \\ y & y & x \\ \\ \end{array}$
Which element acts as the identity for the operation $*$?

128 / 300

For the binary operation table on set $A = \{1, 2\}$ defined by:
$\begin{array}{c|cc} \\ * & 1 & 2 \\ \\ \hline \\ 1 & 2 & 1 \\ \\ 2 & 1 & 2 \\ \\ \end{array}$
Is the operation $*$ commutative?

129 / 300

Given the following binary operation table on set $S = \{a, b\}$:
$\begin{array}{c|cc} \\ * & a & b \\ \\ \hline \\ a & a & b \\ \\ b & b & a \\ \\ \end{array}$
What is $a * b$?

130 / 300

Let $S = \{1, -1, i, -i\}$ where $i$ is the imaginary unit. Under standard multiplication, which element is its own inverse and also the identity element?

131 / 300

Consider the set of rational numbers $\mathbb{Q}$ under multiplication. Which of the following elements does NOT have an inverse in $\mathbb{Q}$?

132 / 300

Let $*$ be a binary operation on the set of real numbers defined by $a * b = a + b + ab$. If the identity element for this operation is $-1$, what is the inverse of $4$?

133 / 300

Let $A = \{1, -1, i, -i\}$ with the operation of multiplication. Which element of $A$ is its own inverse?

134 / 300

Consider the set of rational numbers $\mathbb{Q}$ under multiplication. Which of the following elements does not have a multiplicative inverse in $\mathbb{Q}$?

135 / 300

Let $*$ be a binary operation on the set of real numbers defined as $a * b = a + b - 5$. What is the inverse of $3$ under this operation?

136 / 300

Does the natural number 5 have a multiplicative inverse within the set of natural numbers?

137 / 300

Under multiplication, what is the inverse of $\frac{2}{3}$ in the set of non-zero rational numbers?

138 / 300

What is the additive inverse of 7 in the set of real numbers?

139 / 300

Consider the power set $\mathcal{P}(X)$ of a non-empty set $X$ under the operation of symmetric difference $\Delta$ (defined as $A \Delta B = (A \cup B) \setminus (A \cap B)$). What is the identity element for this operation?

140 / 300

Let $S = \{a, b, c\}$ and define a binary operation $*$ on $S$ by the following table:
$\begin{array}{c|ccc} \\ * & a & b & c \\ \\ \hline \\ a & a & b & c \\ \\ b & b & a & c \\ \\ c & c & c & c \\ \\ \end{array}$
What is the identity element in $(S, *)$?

141 / 300

Consider the set of $2 \times 2$ invertible matrices under matrix multiplication. If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and its inverse is $A^{-1} = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$, what is the result of $A \times A^{-1}$?

142 / 300

Let $S$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ under function composition $\circ$. Which function is the identity element for this operation?

143 / 300

Consider the set of $2 \times 2$ real matrices under matrix multiplication. What is the identity element for this operation?

144 / 300

Let $S = \{a, b, c\}$ be a set with a binary operation $*$ defined by the following table:
\begin{center}
\begin{tabular}{|c|c c c|}
\hline
$*$ & $a$ & $b$ & $c$

\hline
$a$ & $a$ & $b$ & $c$

$b$ & $b$ & $a$ & $b$

$c$ & $c$ & $b$ & $a$

\hline
\end{tabular}
\end{center}
What is the identity element of $(S, *)$?

145 / 300

What is the identity element for the intersection operation ($\cap$) on subsets of a universal set $U$?

146 / 300

Which of the following is the identity element for multiplication of rational numbers?

147 / 300

What is the identity element for the operation of addition on the set of real numbers?

148 / 300

Define the binary operations $*$ and $\circ$ on $\mathbb{R}$ as:
$$ a * b = |a - b| $$
$$ a \circ b = \frac{a + b}{2} $$
Determine whether $*$ is distributive over $\circ$.

149 / 300

Consider the set $A = \{1, 2, 3\}$. Let $B = \{1, 2\}$ and $C = \{2, 3\}$. Verify whether intersection ($\cap$) is distributive over union ($\cup$) for these sets.

150 / 300

Let $A$ be the set of real numbers and define two binary operations $*$ and $\circ$ on $A$ as follows:
$$ a * b = \max(a, b) $$
$$ a \circ b = \min(a, b) $$
Check whether $*$ is distributive over $\circ$.

151 / 300

Let $S$ be the set of all real-valued functions defined on $\mathbb{R}$. Is the composition of functions, denoted by $\circ$, associative on $S$?

152 / 300

Let $M_{2}(\mathbb{R})$ denote the set of $2 \times 2$ real matrices. Is matrix multiplication associative on $M_{2}(\mathbb{R})$?

153 / 300

Consider a binary operation $*$ defined on the set of real numbers as $a * b = 2a + 3b$. Which of the following statements correctly describes the associativity of $*$?

154 / 300

Which of the following operations is NOT a binary operation on the set $T = \{1, 2, 3, 4\}$?

155 / 300

Let $A$, $B$, and $C$ be subsets of a universal set $U$. Which of the following represents the distributive property of intersection over union?

156 / 300

Consider a set $S = \{a, b, c\}$ with a binary operation $\ast$ defined by the following table:
$\begin{array}{c|ccc} \\ \ast & a & b & c \\ \\ \hline \\ a & a & b & c \\ \\ b & b & a & c \\ \\ c & c & c & a \\ \\ \end{array}$
Is the operation $\ast$ commutative and associative?

157 / 300

Let $f(x) = x^2$ and $g(x) = x + 3$ be two functions from $\mathbb{R}$ to $\mathbb{R}$. Is the composition of functions $f \circ g$ equal to $g \circ f$?

158 / 300

Let $A$ and $B$ be two $2 \times 2$ matrices. Is matrix multiplication commutative for these matrices?

159 / 300

Consider the binary operation $*$ defined on the set of rational numbers $\mathbb{Q}$ by $a * b = \frac{a + b}{2}$. Determine whether this operation is commutative.

160 / 300

Let $*$ be a binary operation on $\mathbb{Z}^+$ defined by $a * b = \max(a, b)$. Is $*$ associative?

161 / 300

Consider the binary operations $*$ and $\circ$ on $\mathbb{R}$ defined by $a * b = a + b - ab$ and $a \circ b = a + b$. Does $*$ distribute over $\circ$?

162 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = 3a + 2b$. Which of the following statements is true?

163 / 300

Given the binary operations $*$ and $\circ$ on $\mathbb{R}$ defined by $a * b = 2a - b$ and $a \circ b = a + b$. Verify whether $*$ is distributive over $\circ$.

164 / 300

Consider binary operations $*$ and $\circ$ defined on $\mathbb{R}$ as $a * b = a + b$ and $a \circ b = a \cdot b$. Which statement is true regarding distributivity?

165 / 300

Let $*$ and $\circ$ be binary operations on a non-empty set $A$. For the operation $*$ to be distributive over $\circ$, which of the following must hold for all $a, b, c \in A$?

166 / 300

For the operation $\star$ defined on $\mathbb{Z}$ as $a \star b = a + b + ab$, determine whether it is associative.

167 / 300

Consider the operation $\circ$ on $\mathbb{Q}$ defined by $a \circ b = \frac{a + b}{5}$. Is $\circ$ associative?

168 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $x * y = x + y - xy$. Is the operation $*$ associative?

169 / 300

Given the sets $A = \{1, 2\}$, $B = \{2, 3\}$, and $C = \{3, 4\}$, what is the result of $A \cup (B \cap C)$?

170 / 300

Which of the following operations is NOT commutative on the set of real numbers?

171 / 300

Let $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$, and $C = \{3, 4, 5\}$. What is the result of $A \cap (B \cup C)$?

172 / 300

The binary operation $*$ is defined on the set of natural numbers $\mathbb{N}$ such that $a * b = \text{GCD}(a, b)$. Is $*$ commutative?

173 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = \frac{a - b}{2}$. Is $*$ commutative?

174 / 300

Consider the binary operation $*$ defined on the set of integers $\mathbb{Z}$ by $a * b = a + b - ab$. Is this operation commutative?

175 / 300

Let $*$ be the binary operation defined as $a * b = a^2 + b^2$ and $\circ$ as $a \circ b = ab$. Which of the following holds true for $*$ over $\circ$ on $\mathbb{R}$?

176 / 300

Consider the binary operation $*$ defined on $\mathbb{R}$ by $a * b = 2a + 3b$. Is $*$ associative?

177 / 300

Which of the following operations is commutative on the set of real numbers?

178 / 300

Which of the following correctly represents the distributive property between union ($\cup$) and intersection ($\cap$) of sets?

179 / 300

Which of the following operations is distributive over addition in the set of real numbers?

180 / 300

If $*$ is defined as $a * b = a + b - ab$ and $\circ$ is defined as $a \circ b = a^2 + b^2$, check if $*$ is distributive over $\circ$ for all real numbers $a, b, c$.

181 / 300

Which of the following is an example of a non-associative operation?

182 / 300

Let $*$ be defined on $\mathbb{Q}$ by $a * b = 2ab$. Is $*$ associative?

183 / 300

Which of the following operations is associative on the set of real numbers?

184 / 300

For sets $A$, $B$, and $C$, which of the following represents the distributive property of intersection over union?

185 / 300

Which of the following operations is associative?

186 / 300

Which of the following operations is commutative?

187 / 300

Which of the following operations is NOT commutative on the set of integers $\mathbb{Z}$?

188 / 300

Let $*$ be a binary operation on $\mathbb{Z}$ defined by $a * b = a^2 + b^2$. Is $*$ commutative?

189 / 300

Which of the following binary operations is commutative on the set of real numbers?

190 / 300

Let $*$ be defined as $a * b = a + 2b$ for all $a, b \in \mathbb{R}$. Is this operation distributive over addition?

191 / 300

Which binary operation is associative on the set of real numbers?

192 / 300

Which of the following binary operations is commutative?

193 / 300

Let $*$ and $\circ$ be binary operations on $\mathbb{R}$ defined by $a * b = 2ab$ and $a \circ b = a + b - 1$. Which of the following holds?

194 / 300

Consider the set $S = \{x \mid x \in \mathbb{Q}, x > 1\}$ with the operation $*$ defined as $a * b = a + b - ab$. Which statement is true?

195 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = \sqrt{a^2 + b^2}$. Which of the following properties does $*$ satisfy?

196 / 300

Let $S = \{-1, 0, 1\}$. Is subtraction a binary operation on $S$?

197 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = a + 2b$. Is this operation associative?

198 / 300

Consider the binary operation $*$ defined on $\mathbb{Q}$ by $a * b = \frac{ab}{2}$. Which of the following statements is true about this operation?

199 / 300

Which of the following operations distributes over addition on the set of real numbers \$\mathbb{R}\$?

200 / 300

Which of the following binary operations is associative on the set of real numbers \$\mathbb{R}\$?

201 / 300

Which of the following binary operations is commutative on the set of real numbers \$\mathbb{R}\$?

202 / 300

For a set $B$ with 2 elements, how many associative binary operations can be defined on $B$?

203 / 300

Let $A$ be a set with 4 elements. How many commutative binary operations can be defined on $A$?

204 / 300

Consider a set $A$ with 3 elements. How many distinct binary operations can be defined on $A$?

205 / 300

A set $S$ has 4 distinct elements. Find the total number of binary operations possible on $S$.

206 / 300

If a set has 3 elements, how many distinct binary operations can be defined on it?

207 / 300

What is the number of binary operations on a set with 5 elements?

208 / 300

What is the number of binary operations on a set $C$ containing only 1 element?

209 / 300

If a set $B$ has 4 elements, how many binary operations are possible on $B$?

210 / 300

If a set $A$ has 3 elements, what is the total number of binary operations on $A$?

211 / 300

For a set $C = \{0, 1, 2\}$, what percentage of all possible binary operations on $C$ would satisfy the condition $f(x,y) = x$ for at least one pair $(x,y) \in C \times C$ but not for all pairs?

212 / 300

Let $P = \{\emptyset, \{1\}\}$ be a set representing a power set. What is the number of binary operations on $P$ where the operation applied to any two elements yields the same result as their set union?

213 / 300

Consider a set $B = \{x, y, z\}$. If we define a new binary operation $\star$ that maps $(a,b)$ to the first element of the ordered pair, how many distinct binary operations of this specific type exist on $B$?

214 / 300

Let $S = \{n \mid n \text{ is a divisor of } 12\}$. Define a binary operation $*$ on $S$ by $a * b = \text{LCM}(a, b)$. Is $*$ a valid binary operation on $S$?

215 / 300

Consider a binary operation $\circ$ defined on the set $S = \{1, 2, 3, 4\}$ where $a \circ b = (a + b) \mod 5$. Which statement correctly describes this operation?

216 / 300

Let $S = \{x \mid x \text{ is a prime number less than 10}\}$. Define a binary operation $*$ on $S$ such that $a * b = \text{GCD}(a, b)$. Which of the following is true about this operation?

217 / 300

Let $A = \{x, y, z\}$ be a set with 3 distinct elements. How many different commutative binary operations can be defined on A?

218 / 300

Consider the binary operation $*$ on $\mathbb{R} \setminus \{1\}$ defined by $a * b = a + b - ab$. For an element $a \neq 1$, its inverse with respect to this operation is:

219 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = a + b - ab$. Which of the following statements is true about this operation?

220 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $x * y = xy + x + y$. Which of the following sets is closed under $*$?

221 / 300

Given the set $A = \{[(x, y), p] \mid x, y \in \mathbb{Z}, p \in \mathbb{Z}^+, x^2 + y^2 = p\}$. Which of the following is an element of $A$?

222 / 300

Let $S = \{-1, 0, 1\}$. Consider the binary operation $*$ defined by $a * b = a + b - ab$. Which of the following ordered pairs correctly represents this operation?

223 / 300

Consider the binary operation $*$ on $\mathbb{R}$ defined by $a * b = a + b - 2ab$. Which of the following properties does this operation satisfy?

224 / 300

Let $A = \{1, 2, 3\}$. How many different binary operations can be defined on $A$?

225 / 300

Let $*$ be a binary operation on $\mathbb{Q}$ defined by $a * b = a - b + ab$. Which of the following statements is true?

226 / 300

Let $A = \{1, 2, 3\}$. Which of the following defines a valid binary operation on $A$?

227 / 300

Consider a set $B = \{0, 1\}$. Which of the following statements about binary operations on $B$ is correct?

228 / 300

Let $A = \{x, y, z\}$ be a finite set. How many distinct binary operations can be defined on $A$?

229 / 300

Consider the set $S = \{2, 4, 6\}$ and the operation $*$ defined by $a * b = \frac{a + b}{2}$. Is $*$ a binary operation on $S$?

230 / 300

Let $S = \{-1, 0, 1\}$ and define the operation $*$ as $a * b = \text{min}(a, b)$. Which of the following statements is true about $*$?

231 / 300

Let $S = \{0, 1\}$. Which of the following operations is a binary operation on $S$?

232 / 300

For a set $C$ with 5 elements, determine the number of distinct binary operations possible on $C$.

233 / 300

What is the number of binary operations on the set $B = \{x, y, z\}$?

234 / 300

If $A$ is a set with 4 elements, how many binary operations can be defined on $A$?

235 / 300

Let $S = \{[(x, y), z] \mid x, y \in \mathbb{R}, z \in \mathbb{R}, \text{ and } z = x^2 - y^2\}$. What is the value of $z$ in the ordered pair $[(4, 3), z]$?

236 / 300

Consider the binary operation $\circ$ defined by the set $B = \{[(a, b), c] \mid a, b, c \in \mathbb{N} \text{ and } a \times b = c\}$. Which of the following represents the correct mapping for $a = 5$ and $b = 6$ under this operation?

237 / 300

Given the set $A = \{[(x, y), p] \mid x \in \mathbb{Z}, y \in \mathbb{Z}, p \in \mathbb{Z}^+ \text{ and } x + y = p\}$, which of the following is an element of $A$?

238 / 300

Consider the binary operation $*$ defined by $x * y = x - y + xy$ on the set of integers $\mathbb{Z}$. Is this operation associative?

239 / 300

Which of the following binary operations is commutative on the set of real numbers $\mathbb{R}$?

240 / 300

Let $A = \{1, 2\}$. How many binary operations can be defined on the set $A$?

241 / 300

For a set $B$ with 4 elements, how many possible binary operations are there on $B$?

242 / 300

Let $A$ be a set with 3 elements. What is the number of elements in $A \times A$?

243 / 300

If a set $A$ has 2 elements, how many possible binary operations can be defined on $A$?

244 / 300

If $f$ represents the binary operation of multiplication on the set of positive integers, what is $f(3, 4)$?

245 / 300

Let $S = \{1, 2, 3\}$. Which of the following operations is NOT closed on $S$?

246 / 300

Which of the following correctly defines a binary operation on a set $S$?

247 / 300

Which of the following is an associative binary operation?

248 / 300

Which of the following binary operations is commutative?

249 / 300

Let $A = \{1, 2\}$. What is the total number of possible binary operations on $A$?

250 / 300

Given the binary operation defined by the set $A = \{[(x, y), p] \mid x \in Z^*, y \in Z^*, p \in Z^* \text{ and } x + y = p\}$, which statement is true?

251 / 300

For the set $A = \{[(x, y), p] \mid x \in Z, y \in Z, p \in Z^* \text{ and } x \times y = p\}$, which of the following is NOT a valid element?

252 / 300

Which of the following represents a valid element of the set $A = \{[(x, y), p] \mid x \in Z, y \in Z, p \in Z^* \text{ and } x + y = p\}$?

253 / 300

Consider the binary operation $*$ on $\mathbb{Q}$ defined by $a * b = ab + 1$. Is this operation associative?

254 / 300

Which of the following binary operations is commutative?

255 / 300

Let $A = \{1, 2\}$. How many binary operations can be defined on $A$?

256 / 300

Consider the set $A = \{1, 2, 3, 4, 5\}$ with the binary operation $a * b = \text{LCM}\{a, b\}$. Does an identity element exist for this operation?

257 / 300

For the set $A = \{1, 2, 3, 4, 5\}$ with the binary operation $a * b = \text{LCM}\{a, b\}$, which property holds true?

258 / 300

Let $A = \{1, 2, 3, 4, 5\}$ and define a binary operation * on A by $a * b = \text{LCM}\{a, b\}$. Which of the following statements is correct?

259 / 300

Which of the following statements about subtraction on $\mathbb{Z}$ is false?

260 / 300

Let $S = \{x \in \mathbb{Z} \mid x \geq -5\}$. Is subtraction a binary operation on $S$?

261 / 300

Consider the operation $*$ defined by $a * b = a - b$. For which of the following sets is $*$ a binary operation?

262 / 300

Let $T = \{-1, 1\}$. Define a binary operation $\bullet$ on $T$ as $a \bullet b = |a + b|$. Is $\bullet$ a binary operation on $T$?

263 / 300

Consider the operation $\circ$ defined by $a \circ b = a + b - 1$ for all $a, b \in \mathbb{Z}$. Which of the following sets is closed under $\circ$?

264 / 300

Let $S = \{1, 3, 5, \ldots\}$ be the set of odd natural numbers. If $*$ is defined as $a * b = a + b + 2ab$ for all $a, b \in S$, then which of the following statements is true?

265 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = |a|b$. Which of the following properties does $*$ satisfy?

266 / 300

Consider the set $S = \{2, 4, 6, 8,...\}$ of even natural numbers. Which of the following operations is NOT a binary operation on $S$?

267 / 300

Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = a + b + ab$. Which of the following statements is true?

268 / 300

Let $f(x) = 3x + 1$, $g(x) = x^2 - 2$, and $h(x) = \sqrt{x}$. Verify the associativity property $(h \circ (g \circ f))(x) = ((h \circ g) \circ f)(x)$. Which of the following correctly represents both sides of the equation?

269 / 300

Consider a binary operation $*$ defined on the set $S = \{a, b, c\}$ by the following composition table:
$\begin{array}{c|ccc} \\ * & a & b & c \\ \\ \hline \\ a & c & a & b \\ \\ b & a & b & c \\ \\ c & b & c & a \\ \\ \end{array}$
What is the result of $(b * a) * c$?

270 / 300

Given the functions $f(x) = \frac{1}{x - 2}$ and $g(x) = \sqrt{x + 3}$, find $(f \circ g)^{-1}(x)$.

271 / 300

Consider the set $A = \{1, 2, 3, 4, 5\}$ with the binary operation * defined by $a * b = \text{LCM}\{a, b\}$. Which element(s) in $A$ are invertible?

272 / 300

For the set $A = \{1, 2, 3, 4, 5\}$ with the binary operation * defined as $a * b = \text{LCM}\{a, b\}$, what is the identity element?

273 / 300

Let $A = \{1, 2, 3, 4, 5\}$ and the binary operation * be defined by $a * b = \text{LCM}\{a, b\}$. Which of the following properties does * satisfy?

274 / 300

Which of the following sets is closed under subtraction, making subtraction a binary operation on it?

275 / 300

Let ℤ be the set of all integers. Which of the following correctly describes the closure property of subtraction in ℤ?

276 / 300

Consider the natural numbers set ℕ = \{1, 2, 3, ...\}. Which of the following statements is true about subtraction in ℕ?

277 / 300

Which statement is true regarding binary operations on the set $A = \{\sqrt{2}, -\sqrt{2}\}$?

278 / 300

For which of the following sets is addition a binary operation?

279 / 300

Consider the set $S = \{1, 3, 5, 7\}$ under the operation of addition. What can be concluded about addition as a binary operation on $S$?

280 / 300

If $*$ is a binary operation on $\mathbb{Z}$ defined by $a * b = 5a + 2b$, what is the value of $3 * 4$?

281 / 300

Let $A = \{1, 2, 3\}$. How many binary operations can be defined on $A$?

282 / 300

Which of the following statements is true about binary operations on the set of even natural numbers?

283 / 300

Given $f(x) = x^2$, $g(x) = 2x + 1$, and $h(x) = \sqrt{x}$, what is $(h \circ (g \circ f))(4)$?

284 / 300

If $f(x) = 5x - 7$ and $g(x) = \frac{x + 7}{5}$ are bijective functions, what is $(g \circ f)^{-1}(x)$?

285 / 300

Given $f(x) = 3x + 2$ and $g(x) = \frac{x - 2}{3}$, what is $(g \circ f)(1)$?

286 / 300

What is $(2 * 3) * 4$ when $*$ is defined as $a * b = \text{LCM}\{a, b\}$?

287 / 300

For $a * b = \text{LCM}\{a, b\}$, which statement is true?

288 / 300

What is the value of $3 * 5$ if $a * b = \text{LCM}\{a, b\}$ on the set $A = \{1, 2, 3, 4, 5\}$?

289 / 300

Why is subtraction NOT a binary operation on ℕ but is a binary operation on ℤ?

290 / 300

For which of the following pairs $(a, b)$ does $a - b$ belong to ℤ?

291 / 300

Which of the following examples shows that subtraction is not a binary operation on the set of natural numbers ℕ?

292 / 300

Let $\ast$ be a binary operation on $\mathbb{N}$ given by $a \ast b = \text{HCF}(a, b)$. What is the value of $22 \ast 4$?

293 / 300

Let $\ast$ be a binary operation defined by $a \ast b = a + 3b^2$. What is the value of $2 \ast 4$?

294 / 300

Which of the following operations is a binary operation on the set of odd natural numbers?

295 / 300

Which of the following is NOT a binary operation on the set of odd natural numbers?

296 / 300

Which of the following is an example of a binary operation on the set of even natural numbers?

297 / 300

Is addition a binary operation on the set of even natural numbers?

298 / 300

If binary operation * is defined on set \{a, b\} such that a*a = b and a*b = a, what is the result of a*b?

299 / 300

Which property does composition of functions satisfy?

300 / 300

If $f(x) = x + 2$ and $g(x) = 3x$, what is $(g \circ f)(1)$?

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