Question

Let A be the set of all triangles in the Euclidean plane and R is the relation on A Defined
by ‘a is similar to b’. Then show that R is an equivalence relation on A

Answer 1

Answer

To show that, R is an equivalence relation on A , we need to show that

the Reltn is

1. Reflexive Reltn

2. Symmetric Reltn

3. Transitive Relatn

→→R is the relatn on A Defined  by ‘a is similar to b’, where A is the set of all triangles in the Euclidean plane.

 →→(a~a)→Since Every Triangle is similar to itself.So it is a Reflexive relatn. Denoted by (a,a),which means triangle a is similar to itself.

→→→[a~b→b~a ],which means,if  a is similar to b, then b is similar to a, that is if triangle A is similar to triangle B,then triangle B is similar to triangle A.

So,we conclude that , relatn is symmetric that is ,(a,b)→(b,a).

→→→→If , (a~b), (b~c)→(a~c),triangle A, is similar to triangle B, triangle B is similar to triangle C, which implies that , triangle A is similar to triangle C.Can be Written as, (a,b), (b,c)→(a,c)

So, R has three elements a,b and c,which can be represented by three triangles, then

If , R={a,b,c}

R×R={(a,a),(b,b),(c,c),(a,b)(b,a),(a,c),(c,a),(b,c),(c,b)}

Showing that R is an Equivalence relatn.