previous year question papers gate- GATE 1995-relation-Let R be a symmetric and transitive relation.

Ali Seron Jason

2 months ago

Contents

Question:
Let R be a symmetric and transitive relation on a set A. Then which of the following is always true?

(A) R is reflexive
(B) R is an equivalence relation
(C) R is anti-symmetric
(D) R is reflexive or empty

We are given that R is symmetric and transitive. Let’s analyze the given options:

A relation R is reflexive if (a, a) ∈ R for all a ∈ A.
R is only symmetric and transitive, but there is no guarantee that it contains (a, a) for all elements.
Counterexample: If R = ∅ (empty set), it is still symmetric and transitive but not reflexive.
So, R is NOT necessarily reflexive → (Option A is False).

A relation is an equivalence relation if it is reflexive, symmetric, and transitive.
Since R is given as symmetric and transitive but NOT necessarily reflexive, it is not always an equivalence relation.
Option B is False.

A relation R is anti-symmetric if (a, b) ∈ R and (b, a) ∈ R implies a = b.
However, since R is symmetric, we have (a, b) ∈ R ⇒ (b, a) ∈ R, which contradicts anti-symmetry unless R is the identity relation.
Option C is False.

If R is symmetric and transitive, the only possible cases are:
R is reflexive (if it contains all (a, a) pairs).
R is empty (which still satisfies symmetry and transitivity).
Option D is True (Correct Answer).

Option (D) R is reflexive or empty

This is a common GATE question based on properties of relations. Would you like more practice questions on this topic?