First slide

Relations XII

Question

$Let A = {1, 2, 3} . Then find the number of relations containing (1,2) and (1, 3) which are reflexive and symmetric but not transitive.$

Easy

A

2
B

1
C

3
D

0

Solution

Let relation R be reflexive, symmetric but not transitive containing (1,2) and (1, 3).
Since R is reflexive, we must have
$(1, 1), (2, 2) and (3, 3) \in R$
Since R is symmetric, we must have
$(2, 1), (3, 1) \in R$
Now, if we add one of the two pairs (3, 2) and (2, 3) (or both) to relation R, then relation R will become transitive. Hence, the total number of desired relations is 1.

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