Algebra

Relations & Functions

Medium

The Relation $R = \{(x, y) : x \leq y^2\}$ defined on the set $\mathbf{R}$ of Real numbers is :

(A) reflexive but not symmetric

(B) neither reflexive nor symmetric

(C) neither reflexive nor transitive

(D) reflexive but not transitive

(E) not reflexive but symmetric

Choose the correct answer from the options given below :

(A) and (C) only

(B) and (C) only

(A) and (D) only

✅ Correct Option: 2

Take $x=0.5$ : $0.5 \leq 0.25$ is false, so not reflexive. Take $(2,3)$ : $2\leq 9$ true but $3\leq 4$ true; check $(0.5,1)$ : $0.5\leq 1$ true, $(1,0.5)$ : $1\leq 0.25$ false, not symmetric. $(2,-3)\in R$ : $2\leq 9$ , $(-3,1)$ : $-3\leq 1$ , but $(2,1)$ : $2\leq 1$ false, not transitive. So neither reflexive nor symmetric (B), and neither reflexive nor transitive (C).

2025: 27 May Shift 1

The function $f: \mathbb{R} \rightarrow [-1, 1]$ defined by $f(x) = \cos x$ is:

2025: 22 May Shift 1

A and B are two sets such that $n(A) = 5$ and $n(B) = 7$ . The number of one-one functions from A to B is

2025: 21 May Shift 1

If R be a relation on the set of integers Z, given by R = {(a, b) : (a - b) is a multiple of 3}, then R is:

2025: 16 May Shift 1

If R and S are two equivalence relations on a set A, then

2025: 16 May Shift 1

Match List-I with List-II

Where ℝ is set of real numbers

List-I	List-II
(A) f: ℝ → ℝ s.t f(x) = x⁴ is	(I) one-one, Into
(B) f: ℝ → [0, ∞) s.t f(x) = x⁴ is	(II) many-one, into
(C) f: [0, ∞) → ℝ s.t f(x) = x⁴ is	(III) one-one, onto
(D) f: [0, ∞) → [0, ∞) s.t f(x) = x⁴ is	(IV) many-one, onto

Choose the correct answer from the options given below:

2025: 26 May Shift 2

The value of $\left|\begin{array}{cc}\log_5 10 & 2 \\[4pt] 2 & \log_{10} 5\end{array}\right|$ is

Solution

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