Please Show step by step Thank you A binary relation R is de

Please Show step by step, Thank you

A binary relation R is defined on the set Z^2 as follows: Prove that R is an equivalence relation on Z^2. Determine the equivalence class of (1, 2) with respect to R. Describe it as precisely as possible. How many distinct equivalence classes with respect to R are there in total? Briefly justify your answer.

Solution

a)

Transitivity

(a,b)R(c,d), (c,d)R(e,f)

Hence, a=c=e (mod 2), b=d=f(mod 3)

Hence, (a,b)R(e,f)

So R is transitive

Symmetry

a=a(mod 2) , b=b (mod 3)

So trivially, (a,b)R(a,b)

Hence relation is symmetric.

Reflexive

(a,b)R(c,d) implies a=b mod 2 is b=a (mod 2)

c=d mod 3 ie d=c mod 3

Hence, (c,d)R(a,b)

Hence R is reflexive.

This proves R is an equivalence relation.

b)

Let (a,b) so that: (1,2)R(a,b)

Hence, a=1 mod 2 and , b=2 mod 3

Hence equivalence class of (1,2) is set of elements: (2m+1,3n+2) , where m and n are integers.

c)

So we can see that elements belong to one equivalence class if their first element are equal modulo 2 and second element are equal modulo 3

There are two remainders modulo 2 ie 0 and 1. There are three remainders modulo 3:0,1,2.

First and second element can be varied independently.

Hence total number of equivalence classes is:2*3=6