Question

let P be the set of all triangles in a plane and R be the relation defined on P as aRb if a is similar to b. prove that R is an equivalence relation

Answer 1

A relation R in a set P is an equivalence relation if it is Symmetric, Reflexive as well as Transitive.

1) Symmetricity :
Let there be two triangle T(1) and T(2) in plane B which satisfies the relation such that, T(1) is similar T(2) .
=> We know, If T(1) is similar to T(2) .
=> T(2) is similar to T(1)
=> This relation is symmetric.

2) Relfexitivity :
We know, Every Triangle is similar to itself.
=> (T, T) satisfies the relation on similarity for all triangles in Set P.
= > Given Equation is Reflexive.

3) Transitivity :
Let there be any three triangles in plane P, such that ( T(1) , T(2) ) & ( T(2) , T(3) ) satisfies the relation .
=> T(1) is similar to T(2) & T(2) is similar to T(3) .
=> T(1) is similar to T(3)
=> (T(1),T(3)) also satisfies the relation.
=> Given Relation is transitive.

Since, Given Relation is transitive, Reflexive, and Symmetric.
=€ Given Relation is Equivalence Relation.