Relation properties with examples

(i) R is reflexive if aRa (ii) R is transitive if aRb and bRc implies aRc(i) R is reflexive if aRa (ii) R is transitive if aRb and bRc implies aRc

(iii) R is symmetric if aRb implies bRa. A relation which is reflexive, transitive, and symmetric is called an equivalence relation.

Example 9:-

A relation R defined on the set of integers is defined as R = {(a, b) : a - b is an integer}.

Show that R is an equivalence relation.

Suggested Answer:-

(i) For reflexive property:

Let a - a = 0 is an integer True.

(ii) For symmetric property:

a - b is an integer b - a = -(a - b) is also an integer True.

(iii) For transitive property:

Let a, b c be integers then (a - b) is an integer, (b - c) is an integer.

a - c = (a - b) + (b - c) is also an integer True.

From (i), (ii) and (iii) R is an equivalence relation.

Example 10:-

Show that 'is a brother of' on a set of people is not an equivalence relation.

Suggested Answer:-

1. For reflexive property: aRa for example, x 'is brother of' himself. False

2. For transitive property : Let a, b, c be three persons. If aRb, bRc then aRc. a is brother of b, and b is brother of c then a is brother of c. True.

3. For symmetric property: If aRb then bRa. Let a be brother of b then b may not be brother of a, because b may be sister of a. False.

Note:-

When any one property is not satisfied we say that the relation is not an equivalence relation.

Example 11:-

Let A = {members of a family}. The relation R means 'is the father of'. Show that the relation R does not satisfy any property of relation.

Suggested Answer:-

(i) For reflexive property: aRa a is the father of a,

a is the father of himself. False

(ii) For symmetric property : aRb then bRa.

If a is the father of b then b is the father of a. False.

(iii) For transitive property : aRb, bRc then aRc. If a is the father of b and b is the father of c, then a is a father of c.

It is not possible as a is the grandfather of c. False.

None of the properties are satisfied.