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.