Let W denote the words in the English dictionary. Define the relation R by : R=\ (x, y) ∈ W × W the words x and y have at least one letter in common . Then R is

Correct answer: B. reflexive, symmetric and not transitive Clearly (x, x) ∈ R x ∈ W . So, R is reflexive. Let (x, y) ∈ R , then (y, x) ∈ R as x and y have at least one letter in common. So, R is symmetric. But R is not transitive for example Let x= DELHI, y= DWARKA and z= PARK then (x, y) ∈ R and (y, z) ∈ R but (x, z) R .

JEE Main 2006

Exam: JEE Main
Year: 2006
Subject: Mathematics
Chapter: Sets and Relations

JEE Main 2006 · Mathematics · Sets and Relations

Let W denote the words in the English dictionary. Define the...

Let W denote the words in the English dictionary. Define the relation R by :

R=\{(x, y) \in W \times W \mid

the words

x and

have at least one letter in common

. Then

R

Options

Anot reflexive, symmetric and transitive
Breflexive, symmetric and not transitive
Creflexive, symmetric and transitive
Dreflexive, not symmetric and transitive

Correct answer

B. reflexive, symmetric and not transitive

Step-by-step solution

Clearly

(\mathrm{x}, \mathrm{x}) \in \mathrm{R} \forall \mathrm{x} \in \mathrm{W}

. So,

\mathrm{R}

is reflexive. Let

(x, y) \in R

, then

(y, x) \in R

x

and

y

have at least one letter in common. So,

R

is symmetric. But

R

is not transitive for example Let

x=

DELHI,

y=

DWARKA and

z=

PARK then

(x, y) \in R

and

(y, z) \in R

but

(x, z) \notin R

Practice Sets and Relations on Quantrex Academy →

More from Sets and Relations

Let S = \ 1, 2, 3, …, 10\ . Consider the set X = \ R : R is an equivalence relation on… JEE Advanced 2026 Consider the relation R on the set \ -2,-1,0,1,2\ defined by (a, b) ∈ R if and only if… JEE Main 2026 Let R = \ (x, y) ∈ ℕ × ℕ : _e(x + y) ≤ 2\ . Then the minimum number of elements, required… JEE Main 2026 Let A = \ 1, 4, 7\ and B = \ 2, 3, 8\ . Then the number of elements, in the relation R =… JEE Main 2026 Let A = \ 2, 3, 4, 5, 6\ . Let R be a relation on the set A × A given by (x, y) R (z, w)… JEE Main 2026 Let R be a relation defined on the set \ 1,2,3,4\ ×\ 1,2,3,4\ by R=\ ((a, b),(c, d)): 2… JEE Main 2026 Let A=\ 0,1,2, …, 9\ . Let R be a relation on A defined by (x, y) ∈ R if and only if… JEE Main 2026 Let A=\ -2,-1,0,1,2,3,4\ . Let R be a relation on A defined by x Ry if and only if 2 x+y… JEE Main 2026 The number of elements in the relation R=\ (x, y): 4 x^ 2 +y^ 2 <52, x, y ∈ Z\ is JEE Main 2026 Let S be the set of the first 11 natural numbers. Then the number of elements in A=\ B ⊆… JEE Main 2026 Let the relation R on the set M=\ 1,2,3, …, 16\ be given by R=\ (x, y): 4 y=5 x-3, x, y ∈… JEE Main 2026 Let A=\ x:|x^ 2 -10| ≤ 6\ and B=\ x:|x-2|>1\ . Then JEE Main 2026 Let A=\ 2,3,5,7,9\ . Let R be the relation on A defined by x R y if and only if 2 x ≤ 3 y… JEE Main 2026 The number of relations, defined on the set \ a, b, c, d\ , which are both reflexive and… JEE Main 2026 Let the set of all relations R on the set \ a, b, c, d, e, f\ , such that R is reflexive… JEE Advanced 2025 Let A=\ 0,1,2,3,4,5\ . Let R be a relation on A defined by (x, y) ∈ R if and only if max… JEE Main 2025 Full Sets and Relations list Sets and Relations JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs