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

GATE CS – 1995

Here’s a detailed explanation of the GATE 1995 question based on Relations — specifically focusing on Symmetric and Transitive relations.

GATE 1995 – Set Theory / Relations Question

Question:

Let $RR$ be a symmetric and transitive relation on a set $AA$ . Suppose $(a,b)∈R(a, b) \in R$ .
Then which of the following must also be true?

Options (typical structure):
A) $(b,a)∈R(b, a) \in R$
B) $(a,a)∈R(a, a) \in R$
C) $(b,b)∈R(b, b) \in R$
D) All of the above

Solution with Concept:

We are given:

$RR$ is symmetric:
If $(a,b)∈R⇒(b,a)∈R(a, b) \in R \Rightarrow (b, a) \in R$
$RR$ is transitive:
If $(a,b)∈R(a, b) \in R$ and $(b,c)∈R⇒(a,c)∈R(b, c) \in R \Rightarrow (a, c) \in R$

And we are told:
$(a,b)∈R(a, b) \in R$

Step-by-step Reasoning:

From symmetry:
Since $(a,b)∈R(a, b) \in R$ , ⇒ $(b,a)∈R(b, a) \in R$
Now apply transitivity:
- From $(a,b)∈R(a, b) \in R$ and $(b,a)∈R(b, a) \in R$
  ⇒ By transitivity: $(a,a)∈R(a, a) \in R$
Also:
- From $(b,a)∈R(b, a) \in R$ and $(a,b)∈R(a, b) \in R$
  ⇒ $(b,b)∈R(b, b) \in R$

Final Answer:

D) All of the above

GATE Concept Summary:

Property	Definition
Symmetric	$(a,b)∈R⇒(b,a)∈R(a, b) \in R \Rightarrow (b, a) \in R$
Transitive	$(a,b),(b,c)∈R⇒(a,c)∈R(a, b), (b, c) \in R \Rightarrow (a, c) \in R$
Reflexive	$(a,a)∈R∀a∈A(a, a) \in R \forall a \in A$

Note:
Even though reflexivity is not given, due to symmetry + transitivity and the presence of $(a,b)(a, b)$ , we were able to deduce $(a,a)(a, a)$ and $(b,b)(b, b)$ as members of $RR$ .

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GATE 1995 – Set Theory & Relations Question

Step-by-Step Solution

1. Reflexivity Check

2. Equivalence Relation Check

3. Anti-Symmetry Check

4. Reflexive or Empty Check

Final Answer: