Question

Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by x R y if x y ∈ S. Prove that R is an equivalence relation. Determine the equivalence class

Answer 1

Equivalence Relation : A set is said to have equivalence relation R, if for any a, b, c in X it satiesfies following property :-

i) aRa (reflexivity)

ii) if aRb then bRa ( symetric)

iii) if aRb, bRc then aRc (transitivity)

{aRb means a related to b}

Now, coming to question:

Let,   and xRy if xy belongs to S

Then,   for some  

i) Reflexivity

Since,  

So, i. e reflexive

ii) Symetricity

Let, aRb

To prove: bRa

Since, aRb so,

Hence, if aRb then bRa, so it is symetric

iii) Transitivity.

Let, aRb, bRc

To prove : aRc

Since,  

Hence, aRb

Also, his is so obivious.

Thus, relation R defines equivalence relation on S.

Equivalence classes:(simplest meaning is that break given set, such that same nature elements comes in one part)

positive multiple of 2

negative multiple of 2