Question

He relation (a,b) such that "a" and "b" have the same age ,is defined on the sets of all people.Is it equivalent relation?

Answer 1

To show that (a,b) is an equivalent relation we have to prove 3 properties:

1) Reflexivity: for all a the object a has the same age as a. (It's evident)

2) Symmetry: if a has the same age as b, then b has the same age as a. (It's evident)

3) Transitivity: if a has the same age as b, and b has the same age as c, then a has the same age as c.

Let x be an age of a. As the age of b is the same as the age of a, it's equal to x too.

As the age of c is the same as the age of b, it's equal to x also.

Then the ages of a and c both equal to x, and, hence, are the same.

Thus, (a,b) is an equivalent relation.