previous year question papers gate for cse – GATE 1996 Relations Let R be a non empty relation.

The GATE CSE 1996 exam featured a question on relations, specifically:

Question:

Let $R$ be a non-empty relation on a collection of sets, defined by $ARB$ if and only if $A\cap B=\varnothing$. Then, which of the following statements is true?

Options:

A. $R$ is reflexive and transitive
B. $R$ is symmetric and not transitive
C. $R$ is an equivalence relation
D. $R$ is not reflexive and not symmetric

Answer: B. $R$ is symmetric and not transitive

Explanation:

• Reflexivity: A relation $R$ is reflexive if every element is related to itself. For any set $A$, $A\cap A=A$, which is generally not empty unless $A=\varnothing$. Therefore, $R$ is not reflexive.

• Symmetry: If $A\cap B=\varnothing$, then $B\cap A=\varnothing$ as well. Hence, if $ARB$, then $BRA$; thus, $R$ is symmetric.

• Transitivity: Consider sets $A=\{1\}$, $B=\{2\}$, and $C=\{1,2\}$. We have $A\cap B=\varnothing$ and $B\cap C=\varnothing$, so $ARB$ and $BRC$. However, A∩C={1}≠∅A \cap C = \{1\} \neq \emptysetA∩C={1}=∅, so $ARC$ does not hold. Therefore, $R$ is not transitive.

This analysis confirms that option B is correct.

For a detailed walkthrough of this question, you can refer to the following resources:

• GATE CSE 1996 Question 2.2 on GateOverflow

• Set Theory – GATE CSE 1996 Solved Question – YouTube

If you need further assistance or explanations on related topics, feel free to ask!

previous year question papers gate for cse – GATE 1996 Relations Let R be a non empty relation.

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