GATE CS 2021 Set 1 — Question 53
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Engineering Mathematics → Sets & Combinatorics → Sets, Relations & Functions
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Question
A relation R is said to be circular if and together imply . Which of the following options is/are correct?
Correct answer
(C) If a relation S is reflexive and circular, then S is an equivalence relation.
Solution
An equivalence relation must be reflexive, symmetric, and transitive.
1.Option (A): Reflexive and symmetric does not imply transitivity. For example, on set , is reflexive and symmetric but not transitive as but .
2.Option (B): Circular and symmetric implies transitivity (), but not necessarily reflexivity on the entire domain. If , it is circular and symmetric but not reflexive.
3.Option (C): If is reflexive and circular:
- Symmetry: Let . Since is reflexive, holds. Then by circularity. So is symmetric.
- Transitivity: Let . By circularity, holds. Since is symmetric, . So is transitive.
- Since it is also reflexive, is an equivalence relation.
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