Question

Let X = {1, 2 ……7 }and R = { / X-Y is divisible by 3}. Show that R is an equivalence relation. Draw the graph R.

Answer 1

Answer:

Proof below.

Step-by-step explanation:

We need to show that R is an equivalence relation.

1. Checking for Reflexive:

it is reflexive, if aRa

here (a-a)/3 = 0/3 = 0

Therefore, it is reflexive.

2. Checking for Symmetric

It is symmetric if aRb and bRa

Here, if a - b is divisible by 3. Let it be m

then b - a = -(a-b) will be divisible by 3 .  (-m)

Therefore, it is symmetric.

3. Checking for Transitive

It is symmetric if aRb and bRc then aRc

Here, if a - b is divisible by 3. Let it be m

and b - c divisible by 3. Let it be n

therefore, adding the above equations,

a - c is divisible by 3. (by m-n)

Therefore, it is transitive.

As the function is reflexive, symmetric and transitive,

Therefore, it is an equivalence relation.

Hence proved.