DISCRETE STRUCTURES

DISCRETE STRUCTURES

Q. A relation R in {1,2,3,4,5,6} is given by {(1,2),(2,3),(3,4),(4,4),(4,5)}. This relation is :

(A) reflexive

(B) symmetric

(C) transitive

(D) not reflexive, not symmetric, and not transitive

Ans:- D

Explanation:-

i) Reflexive : A relation R on a set A is reflexive if aRa for every a ϵ A, that is, if (a,a) ϵ R for every a ϵ A.

In the above relation, since the set contains the six elements 1,2,3,4,5 and 6. A relation R on A is reflexive if it contains the six pairs (1,1),(2,2),(3,3),(4,4),(5,5) and (6,6). Since the relation only contains (4,4) and not the rest, it is not reflexive.

ii) Symmetric : A relation R on a set A is symmetric if whenever aRb then bRa, that is if whenever (a,b) ϵ R then (b,a) ϵ R. R is not symmetric if there exists a,b ϵ A such that (a,b) ϵ R but (b,a) does not belong to R.

The above relation is not symmetric because (1,2) ϵ R but (2,1) does not belong to R. The same argument holds good for (2,3),(3,4) and (4,5) as well. So, the relation is not symmetric.

iii) Transitive: A relation R on a set A is transitive if whenever aRb and bRc then aRc, that is, whenever (a,b),(b,c) ϵ R then (a,c) ϵ R. The above relation is not transitive because (1,2), (2,3) ϵ R but (1,3) does not belong to R.

S, the above relation is not reflexive, not symmetric, and not transitive.