Prove that R is an equivalence relation. Determine the distinct equivalence classes.

Course: Foundations of Higher Math

A relation is defined on by *x R y* if is even. Prove that is an equivalence relation. Determine the distinct equivalence classes.

Reflexivity:

Let .

which is even, so *x R x.*

Symmetry:

Let .

Assume that *x R y* ,i.e. , .

which is even, so *y R x.*

Is this method fine?

Also, how would I notate the equivalence classes? I believe that there will be 2 distinct classes because 3x and 7y must have same parity for the difference to be even.

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Re: Prove that R is an equivalence relation. Determine the distinct equivalence class

Quote:

Originally Posted by

**MadSoulz** Course: Foundations of Higher Math

A relation

is defined on

by

*x R y* if

is even. Prove that

is an equivalence relation. Determine the distinct equivalence classes.

Reflexivity:

Let

.

which is even, so

*x R x.*

Symmetry:

Let

.

Assume that

*x R y* ,i.e.

,

.

which is even, so

*y R x.*

Is this method fine?

So far but you haven't finished. You still have to prove "transitivity": if x R y and y R z then x R z.

Quote:

Also, how would I notate the equivalence classes? I believe that there will be 2 distinct classes because 3x and 7y must have same parity for the difference to be even.

1.

2.

?

So you are saying that all odd numbers are equivalent and all even numbers are equivalent?

Yes that is correct and your notation is good.

Re: Prove that R is an equivalence relation. Determine the distinct equivalence class