the school has to be less than meters. This shows that R ′ = A × A is the universal relation. Remark In Class XI, we have seen two ways of representing a relation, namely raster method and set builder method. However, a relation R in the set { , , , } defined by R = {( a , b ) : b = a + } is also expressed as a R b if and only if b = a + by many authors.
We may also use this notation, as and when convenient. If ( a , b ) ∈ R, we say that a is related to b and we denote it as a R b . One of the most important relation, which plays a significant role in Mathematics, is an equivalence relation . To study equivalence relation, we first consider three types of relations, namely reflexive, symmetric and transitive.
Definition A relation R in a set A is called (i) reflexive , if ( a , a ) ∈ R, for every a ∈ A, (ii) symmetric , if ( a , a ) ∈ R implies that ( a , a ) ∈ R, for all a , a ∈ A. (iii) transitive , if ( a , a ) ∈ R and ( a , a ) ∈ R implies that ( a , a ) ∈ R, for all a , a , a ∈ A. Definition A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive. Example Let T be the set of all triangles in a plane with R a relation in T given by R = {(T , T ) : T is congruent to T }.
Show that R is an equivalence relation. Solution R is reflexive, since every triangle is congruent to itself. Further, (T , T ) ∈