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

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

https://www.gyanodhan.com/video/7B5.%20GATE%20CSEIT/Discrete%20Mathematics%202/510.%20previous%20year%20question%20papers%20gate%20for%20cse%20-%20GATE%201996%20%20%20Relations%20%20%20Let%20R%20be%20a%20non%20empty%20relation.mp4

In the GATE 1996 Computer Science exam, there was a question regarding a relation $R$ defined on a collection of sets, where $ARB$ if and only if $A\cap B=\varnothing$. The question asked to identify the correct properties of this relation.

Analysis of the Relation $R$:

• 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: A relation $R$ is symmetric if $ARB$ implies $BRA$. Given $A\cap B=\varnothing$, it follows that $B\cap A=\varnothing$. Thus, $R$ is symmetric.

• Transitivity: A relation $R$ is transitive if $ARB$ and $BRC$ imply $ARC$. However, consider $A=\{1\}$, $B=\{2\}$, and $C=\{1,2\}$. Here, $A\cap B=\varnothing$ and $B\cap C=\varnothing$, but A∩C={1}≠∅A \cap C = \{1\} \neq \emptysetA∩C={1}=∅. Therefore, $R$ is not transitive.

Conclusion:

The relation $R$ is symmetric but neither reflexive nor transitive. Therefore, it is not an equivalence relation.

Answer: $R$ is symmetric and not transitive.

For a detailed walkthrough of this problem, you can refer to the following video: