Consider the following relation R on the set of integers ab

Consider the following relation R on the set of integers: (a,b) ? R if and only if 5 | (a-b). Prove that R is reflexive, symmetric, and transitive.

Solution

Reflexive:

5 | (5-5) [0 divided by any number is 0. So, 5 | 0.]

So, (5, 5) belongs to R.

R is reflexive.

Symmetric:

Let (a, b) belongs to R.

Then 5 | (a-b).

If 5 | (a-b) then 5 | -(a-b).

That is 5 | (b-a).

So, (b, a) belongs to R.

R is symmetric.

Transitive:

Let (a, b) belongs to R and (b, c) belongs to R.

Then 5 | (a-b) and 5 | (b-c).

Now, (a - c) = (a - b + b - c)

                  = (a - b) + (b - c)

Since 5 | (a-b) and 5 | (b-c), 5 | (a - b) + (b - c) also.

That is, 5 | (a - c).

R is transitive.