Algebra

Relations & Functions

Medium

core

Let L be the set of all lines in a plane and R be the relation on set L defined by $R = \{(L_1, L_2): L_1 \perp L_2\}$ Then R is

(A) an equivalence Relation

(B) a symmetric Relation

(C) not a transitive Relation

(D) a reflexive Relation

Choose the correct answer from the options given below:

(A) only

(B) and (C) only

(B) and (D) only

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

✅ Correct Option: 2

The set $L$ contains all lines in a plane. The relation $R$ connects two lines if they are perpendicular to each other: $R = \{(L_1, L_2): L_1 \perp L_2\}$ .

To determine the properties of this relation, checking reflexive, symmetric, transitive, and equivalence properties is required.

A line cannot be perpendicular to itself. Two lines are perpendicular only when they meet at 90°, but a line with itself is the same line (angle = 0°).

Therefore, $R$ is not reflexive.

If line $L_1$ is perpendicular to line $L_2$ , then $L_2$ is also perpendicular to $L_1$ . When line 1 makes 90° with line 2, line 2 also makes 90° with line 1.

Therefore, $R$ is symmetric.

Consider three lines where:

$L_2$ is a vertical line
$L_1$ is a horizontal line (perpendicular to $L_2$ )
$L_3$ is also a horizontal line (perpendicular to $L_2$ )

Here $L_1 \perp L_2$ and $L_2 \perp L_3$ , but $L_1$ and $L_3$ are parallel, not perpendicular.

Therefore, $R$ is not transitive.

An equivalence relation requires all three properties: reflexive, symmetric, and transitive.

Since $R$ is not reflexive and not transitive, $R$ is not an equivalence relation.

From the analysis:

(A) Not an equivalence relation
(B) Symmetric relation ✓
(C) Not a transitive relation ✓
(D) Not a reflexive relation

The correct answer is (B) and (C) only.

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:

2023: 23 May Shift 3

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

2022: 7 Aug Shift 2

The modulus function $f : R \to R$ , given by $f(x) = |x|$ , is :

2026: 29 May Shift 2

Match the LIST-I with LIST-II (Consider N and R as set of natural numbers and set of real numbers respectively)

	LIST-I Function		LIST-II
A.	$f: N \to N, f(x) = x^2$ is	I.	Neither one-one nor onto
B.	$f: R \to R, f(x) = 3 - 2x$ is	II.	one-one but not onto
C.	$f: R \to R, f(x) = x^2$ is	III.	onto but not one-one
D.	$f: \{-1,1\} \to \{1\}, f(x) = \vert x \vert$ is	IV.	one-one and 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

2025: 19 May Shift 1

A function $f: \mathbb{R} \rightarrow \{x \in \mathbb{R}: -1 < x < 1\}$ is defined as $f(x) = \frac{x}{1+|x|}$ , then $f$ is:

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