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Is it true that the every relation which is symmetric and transitive is also reflexive ? Give reasons.
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Consider the set I of the integers and a relation be defined as aRb if both a and b are odd.
Clearly aRb bRa i.e. if a and b are both odd then b and a are also both odd. Similarly, aRb and bRc implies aRc and hence transitive. But this relation is not reflexive because 2 I but 2 is not related to 2. In general, any even number is not R-related to itself. Hence it is not reflexive.
Clearly aRb bRa i.e. if a and b are both odd then b and a are also both odd. Similarly, aRb and bRc implies aRc and hence transitive. But this relation is not reflexive because 2 I but 2 is not related to 2. In general, any even number is not R-related to itself. Hence it is not reflexive.
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| Question Text | Is it true that the every relation which is symmetric and transitive is also reflexive ? Give reasons. |
| Answer Type | Text solution:1 |
| Upvotes | 150 |
