Let A 1234 Give an example of a nonempty binary relation on

Let A = {1,2,3,4}. Give an example of a non-empty binary relation on the set A that is: (a) reflexive and symmetric, but not antisymmetric; (b) symmetric and antisymmetric, but not reflexive; (c) transitive but neither symmetric nor antisymmetric; (d) antisymmetric but not transitive.

Solution

(a)

Define a relation R so that aRb if a and b both belong to the set A.

Hence the relation is clearly reflexive ie aRa if a belongs to A

Relation is symmetric because aRb means a and b belong to A and hence bRa

But aRb and bRa does not imply a=b eg. 1R2 and 2R1 in this relation.

Hence the relation is not antisymmetric

(b)

Let R be the relation.

Hence symmetric property requires that if aRb implies bRa

But antisymmetry property requires that if aRb and bRa then a=b hence aRa or bRb

But we require relation to not be reflexive. Hence aRa and bRb is not possible

Hence such a relation is not possible

(c)

Consider the relation: R defined by < is less than

aRb is a<b

This is clearly transitive because

aRb and bRc implies a<b,b<c hence a<c and hence aRc

aRb implies a<b hence b<a is not possible hence R is not symmetric

It is also not antisymmetric because aRb and bRa both are not possible as relation is not reflexive

(d)

Define a relation R such that:

aRb implies: 0<=a-b=1

Hence, 1R1,2R1 and so on.

aRb and bRa implies

a-b>=0 and a-b>=0 hence a=b

2R1 and 3R2 but 3-1 =2 hence 3 is not related to 1 hence relation is not transitive.

Consider the relation R such that aRb i