Algebra

Relations & Functions

Easy

core

Let a function $f:\mathbb{R}\to\mathbb{R}$ defined as $f(x)=x-[x]$ , (where $\mathbb{R}$ is set of real numbers & $[.]$ denotes greatest integer function) .Then the function:

$f(x)$ is one-one as well as onto

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

$f(x)$ is one-one only

$f(x)$ is onto only

✅ Correct Option: 2

$f(x)=x-[x]$ is the fractional part, which is periodic with period $1$ , so $f(0)=f(1)=0$ and $f$ is not one-one.

Its range is $[0,1)$ rather than all of $\mathbb{R}$ , so it is not onto either.

2022: 7 Aug Shift 2

Let L be the set of all lines in a plane and R be the relation in L defined as $R = \{(l_1, l_2) : l_1 \text{ is perpendicular to } l_2, \text{ where } l_1, l_2 \in L\}$ . Choose the correct answer :

2026: 14 May Shift 2

Given the relation $R=\{(1,2),(2,3)\}$ on a set $A=\{1,2,3\}$ , then which of following statements are TRUE?

A. Minimum number of ordered pairs are added to R so that enlarged relation is reflexive is 3

B. Minimum number of ordered pairs are added to R so that enlarged relation is symmetric is 2

C. Minimum number of ordered pairs are added to R so that enlarged relation is an equivalence relation is 7

D. Minimum number of ordered pair are added to R so that enlarged relation is an equivalence relation is 6.

Choose the correct answer from the options given below:

2026: 15 May Shift 1

Consider a function $f: \left[0, \dfrac{\pi}{2}\right] \to R$ given by $f(x) = \sin x$ and $g: \left[0, \dfrac{\pi}{2}\right] \to R$ given by $g(x) = \cos x$ , where $R$ is set of real numbers then which of the following are correct?

A. $f$ is one-one

B. $g$ is one-one

C. $f + g$ is one-one

D. $f + g$ is not one-one

Choose the correct answer from the options given below:

2022: 30 Aug Shift 1

Which of the following is true on the basis of above diagram?

2022: 25 May Shift 1

Let A = {1,2,3}. Consider the relation R = {(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}. Then R is

2026: 22 May Shift 1

Let X be a non empty set and S be the collection of all subsets of X. A relation R in S is defined by $R=\{(A,B): A\subseteq B\}$ , where $X\subseteq Y$ means $X$ is proper subset of $Y$ . Then R is:

Let a function f:R→Rf:\mathbb{R}\to\mathbb{R}f:R→R defined as f(x)=x−[x]f(x)=x-[x]f(x)=x−[x], (where R\mathbb{R}R is set of real numbers & [.][.][.] denotes greatest integer function) .Then the function:

Solution

Related questions:

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CUET Mathematics Past Year Questions (Topic-Wise)

Let a function $f:\mathbb{R}\to\mathbb{R}$ defined as $f(x)=x-[x]$ , (where $\mathbb{R}$ is set of real numbers & $[.]$ denotes greatest integer function) .Then the function: