Algebra

Relations & Functions

Medium

core

Let a relation $R$ be defined on a set $A = \{1,2,3,4,5\}$ as $R = \{(a,b): |a-b| \text{ is even or zero}\}$ . Then which of the following statements is not TRUE?

$R$ is an equivalence relation.

The number of elements in $R$ is 13.

No element of $\{1,3,5\}$ is related to any element of $\{2,4\}$ .

The number of elements in $R$ is 25.

✅ Correct Option: 4

$|a-b|$ even splits $A$ into equivalence classes $\{1,3,5\}$ and $\{2,4\}$ , so $R$ is an equivalence relation and no element of one class relates to the other.

$|R| = 3^2+2^2 = 13$ , so the claim that $R$ has 25 elements is false.

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Let $A$ be set of all real numbers and $R$ be a relation on $A$ defined by $R=\{(a,b): 1+ab>0\}$ , then relation $R$ is

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Match List-I with List-II

Let $f: A \rightarrow B$ be a function given by $f(x) = x^2$

List-I	List-II
Domain and Co-domain	Kind
(A) $A = \mathbb{R}$ and $B = \mathbb{R}$	(I) $f$ is both one-one and onto
(B) $A = \mathbb{R}$ and $B = [0, \infty]$	(II) $f$ is one-one but not onto
(C) $A = B = [0, \infty]$	(III) $f$ is not one-one but onto
(D) $A = [0, \infty]$ and $B = \mathbb{R}$	(IV) $f$ is neither one-one nor onto

Choose the correct answer from the options given below:

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Let $f(x) = x^3$ be a function with domain {0, 1, 2, 3} then domain of $f^{-1}$ is :

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The function $f: [-1, 1] \rightarrow R$ is given by $f(x) = \frac{x}{x + 2}$

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Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = [x]$ , where [x] denotes the greatest integer less than or equal to x. Then which of the following statements are correct?

(A) f is one-one but not onto

(B) f is not onto

(D) f is one-one and onto

Choose the correct answer from the options given below:

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The relation R in $\mathbb{R}$ (set of real numbers) is defined by $R = \{(a,b): a \leq b^3\}$ , then R is

Let a relation RRR be defined on a set A={1,2,3,4,5}A = \{1,2,3,4,5\}A={1,2,3,4,5} as R={(a,b):∣a−b∣ is even or zero}R = \{(a,b): |a-b| \text{ is even or zero}\}R={(a,b):∣a−b∣ is even or zero}. Then which of the following statements is not TRUE?

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Let a relation $R$ be defined on a set $A = \{1,2,3,4,5\}$ as $R = \{(a,b): |a-b| \text{ is even or zero}\}$ . Then which of the following statements is not TRUE?