Please do not answer if you do not understand the problem Sh

Please do not answer if you do not understand the problem. Show all steps and explain clearly. Please type answer or write very clearly. Handwriting can be difficult to make out. Thanks much.

https://d2vlcm61l7u1fs.cloudfront.net/media%2F9cb%2F9cb93720-1f9b-46ba-8295-812a41953fae%2FphpDhFWFn.png

if it looks like this it is completely illegible...i will not be able to make out anything

Solution

a)

1. Denote the relation by R

For any m in Z

m^2-m^2=0 is divisible by 12

Hence, m~m

Hence, R is reflexive

2. Let, m~n

So, m^2-n^2 is divisible by 12

Hence, n^2-m^2 is also divisible by 12

So, n~m

Hence, R is symmetric

3. Let,m~n, n~k

n^2-m^2 is divisible by 12

k^2-n^2 is divisble by 12

(n^2-m^2)+(k^2-n^2) =k^2-m^2 is divisible by 12

So, m~k

Hence, R is transitive

Hence, R is equivalence relation

b.

Not an equivalence relation as it is not transitive

eg.

-1~0 and 0~1

But, -1~1 is not true

Hence not transitive