Algebra

Relations & Functions

Easy

core

The greatest integer function $f: R\to R$ defined by $f(x) = [x]$ , then which of the following are true?

(A) $f(x)$ is one - one and onto.

(B) Domain of $f(x) = R$ (the set of real numbers).

(C) $f(x)$ is neither one-one nor onto.

(D) Range of $f(x) = Z$ (The set of integers).

Choose the correct answer from the options given below:

(A) and (C) only

(B), (C) and (D) only

(A) and (D) only

✅ Correct Option: 2

$[x]$ is defined for every real $x$ , so the domain is $R$ and (B) is true.

All $x$ in $[0,1)$ map to 0, so it is not one-one; its image is $Z$ , a proper subset of the codomain $R$ , so it is not onto. Hence (C) and (D) are true and (A) is false.

2025: 13 May Shift 1

Let $f: \mathbb{R}$ -> $\mathbb{R}$ be a function defined as $f(x) = x^4$ . Which one of the following is true?

2022: 23 Aug Shift 1

If $n(A) = 3$ , $n(B) = 2$ , then number of all possible surjective function from set A to set B are :

2023: 23 May Shift 3

The number of all onto functions from the set {1, 2, .......n} to itself is

2025: 13 May Shift 2

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:

2026: 15 May Shift 1

If $A$ be the set of all real numbers and R be the relation on A defined by $R = \{(a,b): a^2+b^2=1, \forall a,b \in A\}$ , then R is

2026: 29 May Shift 1

Let $R_1, R_2, R_3, R_4$ be the relations defined on the set $A=\{1,2,3\}$ . then

Match List-I with List-II

List-I (Relation)	List-II (Type)
(A) $R_1=\{(1,1),(2,2),(3,3)\}$	(I) Only transitive
(B) $R_2=\{(1,2),(2,3),(1,3)\}$	(II) Equivalence relation
(C) $R_3=\{(1,2),(2,1),(1,1),(2,2)\}$	(III) Reflexive and symmetric but not transitive
(D) $R_4=\{(1,1),(2,2),(3,3),(2,3),(3,2),(1,2),(2,1)\}$	(IV) Symmetric and transitive but not reflexive

Choose the correct answer from the options given below:

Solution

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