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.
In the GATE 1996 Computer Science exam, there was a question regarding a relation RR defined on a collection of sets, where A R BA \, R \, B if and only if A∩B=∅A \cap B = \emptyset. The question asked to identify the correct properties of this relation.
Analysis of the Relation RR:
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Reflexivity: A relation RR is reflexive if every element is related to itself. For any set AA, A∩A=AA \cap A = A, which is generally not empty unless A=∅A = \emptyset. Therefore, RR is not reflexive.
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Symmetry: A relation RR is symmetric if A R BA \, R \, B implies B R AB \, R \, A. Given A∩B=∅A \cap B = \emptyset, it follows that B∩A=∅B \cap A = \emptyset. Thus, RR is symmetric.
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Transitivity: A relation RR is transitive if A R BA \, R \, B and B R CB \, R \, C imply A R CA \, R \, C. However, consider A={1}A = \{1\}, B={2}B = \{2\}, and C={1,2}C = \{1, 2\}. Here, A∩B=∅A \cap B = \emptyset and B∩C=∅B \cap C = \emptyset, but A∩C={1}≠∅A \cap C = \{1\} \neq \emptyset. Therefore, RR is not transitive.
Conclusion:
The relation RR is symmetric but neither reflexive nor transitive. Therefore, it is not an equivalence relation.
Answer: RR is symmetric and not transitive.
For a detailed walkthrough of this problem, you can refer to the following video: