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