Example of Non-transitive relation : Height of Boys R = {( a1 , a2 , a3 ) : Height of a1 is not equal to height of a2 & Height of a2 is not equal to height of a3 à Height of a1 is not equal to height of a3 }
Equivalence Relation
A relation R in a set A is said to be an equivalence relation if R is reflexive , symmetric and transitive
E . g .: Height of Boys R = {( a , a ) : Height of a is equal to height of a }
Set of all triangles in plane with R relation in T given by R = {( T1 , T2 ) : T1 is congruent to T2 }.
Numerical : Show that the relation R in the set { 1 , 2 , 3 } given by R = {( 1 , 1 ), ( 2 , 2 ), ( 3 , 3 ), ( 1 , 2 ), ( 2 , 3 )} is reflexive but neither symmetric nor transitive .
Solution : Since Relation R has elements {( 1 , 1 ), ( 2 , 2 ), ( 3 , 3 )}, so I is Reflexive Relation R has ( 1 , 2 ), but , it doesn ’ t have ( 2,1 ), so it is not symmetric Relation R has ( 1 , 2 ) & ( 2 , 3 ), but it doesn ’ t have ( 1 , 3 ), so it is not transitive
Numerical : Determine if relation is reflexive , symmetric and transitive : Relation R in the set A of human beings in a town at a particular time given by
o R = {( x , y ) : x and y work at the same place } o R = {( x , y ) : x is exactly 7 cm taller than y } Solution : Lets solve for R = {( x , y ) : x and y work at the same place } first .
The relation will have values ( x , x ), ( y , y ) also , since x & x will work at same place . So it is reflexive
If x & y works at same place , then y & x will also work at same place . This relation R will have values ( x , y )( y , x ), so it is Transitive too .
If x & y works at same place , also it y & z works at same place , it implies that x & z works at same place .
Thus relation R will have value ( x , y ), ( y , z ), ( x , z ), so it is transitive too . Thus it is equivalence relation . Let ’ s take case 2 : R = {( x , y ) : x is exactly 7 cm taller than y }, that is x-y = 7 x-x = 0 , not 7 . Thus the relation will not have ( x , x ), so it is not reflexive