Prove that the following are an equivalence relation Reflexi

Prove that the following are an equivalence relation. (Reflexive, Symmetric, and Transitive)

Solution

Reflexive property

a~a because a-a=0 is a multiple of 5.

Therefore ~ is a Reflexive relation.

Symmetric property

Let a~b

=> a-b is a multiple of 5

=> -(b-a) is a multiple of 5

=> b-a is a multiple of 5

=> b~a

Therefore ~ is a Symmetric relation.

Transitive property

Let a~b and b~c

=> a-b and b-c is multiple of 5.

=> a-b+b-c is multiple of 5.

=> a-c is multiple of 5.

=> a~c

Therefore ~ is a Transitive relation.

Therefore ~ is an equivalence relation. (Proved )