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

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

GATE 1995 – Set Theory & Relations Question

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

 Step-by-Step Solution

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

 1. Reflexivity Check

• 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).

 2. Equivalence Relation Check

• 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.

 3. Anti-Symmetry Check

• 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.

 4. Reflexive or Empty Check

• 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).

 Final 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?

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:

1. From symmetry:
   Since (a,b)∈R(a, b) \in R, ⇒ (b,a)∈R(b, a) \in R
2. 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
3. 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:

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.

Would you like a PDF of:

• All GATE relation-based questions (1991–2024)?
• Concept revision sheet (Properties of Relations)?
• Practice questions with Venn diagrams?

Let me know and I’ll prepare it for you!

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