Answer to Question #180701 in Discrete Mathematics for Durjoy

Question #180701

Draw the digraph and the matrix of the relation R= {(1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3,4), (4, 1), (4, 2), (4, 3)} on the set A= {1, 2, 3, 4, 5}. Also decide whether it is reflexive,

whether it is symmetric, whether it is anti symmetric,whether it is transitive.



1
Expert's answer
2021-04-14T10:51:50-0400


Above is the digraph for the relation

The relation is not refexive. This is because according to definition of reflexive, "aRa \\forall a\\in A" but "3\\not R 3, 4 \\not R4" etc. Hence the relation is not reflexive.

Also, the relation is not symmetric. By definition of symmetric, if "aRb" ,then "bRa \\forall a,b \\in A" . But, "2R3" and "3 \\not R 2."

Hence, the relation is not symmetric.

For anti-symmetry, if "aRb" and "bRa \\implies a=b \\forall a,b \\in A" . 3R4 and 4R3 but, "3 \\neq 4." Hence the relation is not anti-symmetric.

For transitive, if "aRb" and "bRc," then "aRc \\forall a,b,c\\in A" . 1R3 and 3R4 ,but "1 \\not R 4" . Hence the relation is not transitive.


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