discrete math question problem helo please with brief soluti

discrete math question problem helo please with brief solution with steps and explanation

   For a ,b Z, define a ~ b if and only if a- b is divisible by 3.
   Prove that ~ defines an equivalence relation on Z.
   What is [0]?
   What is [1]?

Solution

1. Check if ~ is reflexive

a-a=0 is divisible by 3 hence a~a

ie ~ is reflexive

2. Check if ~ is transitive

Let a~b and b~c

SO ,a-b is divisible by 3 and b-c is divisible by 3

Hence, a-b+b-c is divisible by 3

But, a-b+b-c=a-c

So, a~c

Hence, ~ is transitive

3. Check if ~ is symmetric

a-b divisible by 3 means b-a divisible by 3

hence, ~ is symmetric.

So , ~ is an equivalence relation

[0] contains all numbers so that:a-0 is divisible by 3 ie all multiples of 3

[1] contains all numbers so that:a-1 is divisible by 3 ie all numbers of the form:3k+1