Properties of Relations

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Consider the set A = {a, b, c, …} then any subset of A x A is called a relation in A. Let R be any relation in A so that R is a subset of the product set A x A. We will discuss the properties a relation may possess.Consider the set A = {a, b, c, …} then any subset of A x A is called a relation in A. Let R be any relation in A so that R is a subset of the product set A x A. We will discuss the properties a relation may possess.

Reflexive property

A relation R in a set A is said to be reflexive if every element of the set A is related to itself. This is if aRa where a Î A, or (a, a) Î R for each a Î A

For example,

(i) a triangle 'is congruent to' itself.

(ii) 5 is a multiple of' 5.

Transitive property

A relation R, is said to be transitive if a 'is related to' b and b 'is related to' c then a 'is related to' c, (a, b) R and if aRb, bRc then aRc.

For example,

(i) If  DABC 'is congruent to' DDEF and DDEF 'is congruent to' DXYZ, then DABC 'is congruent to' DXYZ

(ii) If Ram 'is the brother of' Laxman and Laxman 'is the brother of' Bharat then Ram 'is the brother of' Bharat.

Symmetric property

A relation R, on a set A, is symmetric if aRb then bRa if a 'is related to' b, b 'is related to' a for a, b Î R. Also if   (a, b) Î R then (b, a) Î R

For example,

(i) If DABC 'is congruent to' DXYZ, then DXYZ 'is congruent to' DABC.

(ii) If Mary 'is the sister of' Lucy, then Lucy 'is the sister of' Mary.

Equivalence Relation

If a relation is (i) reflexive, (ii) transitive and, (iii) symmetric, then it is called an equivalence relation.

For example,

(i) is congruent to (ii) is parallel to (iii) is similar to, are some examples of equivalence relations.