Question

Given that R = {(a, b) (3 divides a - b) is an equivalence relation in the set of integer Z what is the number of partitions of Z ?​

Answer 1

Answer:

Step-by-step explanation:

Given that:

R={(a, b) : 3 divides a−b}

Case I : Reflexive

Since,  a−a=0

And                                              

3  divides  0,   ∴    

0/2  

​ =0

⇒2 divides  a−a

∴(a, 0) ∈ R.

∴R is reflexive

Case II : - Symmetric

If 3 divides a - b,

then, 3 divides - (a - b) i.e. b - a

 Hence, If ( a , b) ∈R, Then ( b , a) ∈R

 ∴R is symmetric.

Case III. Transitive.

 If 3 divides ( a - b) and 3 divides ( b - c)  

So, 2 divides ( a - b) + ( b - c)

 also,

 2 divides ( a - c)

 ∴If(a,b) ∈R and (b, c) ∈R

 then,

 (a, c) ∈R

 Thus, R is an equivalence relation in z.