# Thread: Equivalence relations X x X

1. ## Equivalence relations X x X

Let X ={1,2, ...,10} Define a relation R on X x X by (a,b) R (c,d) if a+d=b+c
How can I show that R is an equivalence relation on X x X?
Thank you!

2. ## Re: Equivalence relations X x X

What's the definition of an equivalence relation R on X? (or even better: for which conditions is a relation an equivalence relation?)

3. ## Re: Equivalence relations X x X

A relation that is reflexive, symmetric, and transitive on a set is called an equivalence relation.

4. ## Re: Equivalence relations X x X

Originally Posted by mathproblems
A relation that is reflexive, symmetric, and transitive on a set is called an equivalence relation.
Indeed! So use them to prove that R is an equivalence relation on X.

5. ## Re: Equivalence relations X x X

I am not sure how to start, and how to use a+d=b+c
Do I plug numbers between 1 to 10 in there?

Let (a,b) R (c,d) be symmetric
therefore R { (a,b) , (c,d) ,(b,a), (d,c)}

Let (a,b) R (c,d) be reflexive
therefore R {(a,b), (c,d)}

Let (a,b) R (c,d) be transitive
therefore R {(a,b) (b,c) (c,d)}

So the Equivalence relation of (a,b) R (c,d) will be {(a,b) , (c,d) ,(b,a), (d,c) (b,c)}

where shall I use a+d= b+c?
Thank you.

6. ## Re: Equivalence relations X x X

Originally Posted by mathproblems
I am not sure how to start, and how to use a+d=b+c
Do I plug numbers between 1 to 10 in there?
Let (a,b) R (c,d) be symmetric
therefore R { (a,b) , (c,d) ,(b,a), (d,c)}
Let (a,b) R (c,d) be reflexive
therefore R {(a,b), (c,d)}
Let (a,b) R (c,d) be transitive
therefore R {(a,b) (b,c) (c,d)}
So the Equivalence relation of (a,b) R (c,d) will be {(a,b) , (c,d) ,(b,a), (d,c) (b,c)}
where shall I use a+d= b+c?
Once again, that sort of example has absolutely nothing to do with proof.
Do you understand what constitutes a proof?

I will get you started.
We know that $a+b=b+a$ therefore $(a,b)\mathbf{R}(a,b)$.
Therefore, $\mathbf{R}$ is reflexive.

7. ## Re: Equivalence relations X x X

could you give me a hint on the numbers, how do I use them a+d=b+c? Thank you.

8. ## Re: Equivalence relations X x X

Originally Posted by mathproblems
could you give me a hint on the numbers, how do I use them a+d=b+c? Thank you.
Numbers have nothing to do with proof.
Did you understand the proof the $\mathbf{R}$ is reflexive?

Now suppose $(a,b)\mathbf{R}(c,d)$.
To prove symmetry we must show that $(c,d)\mathbf{R}(a,b)$.

We know that $(a,b)\mathbf{R}(c,d)$ means that $a+d=b+c$.
But we can rearrange that to get $c+b=d+a$.
That means $(c,d)\mathbf{R}(a,b)$.
Thus it is symmetric.

9. ## Re: Equivalence relations X x X

thank you. i understood.