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## Homework Statement

Given: A relation R over N x N

((x,y), (u,v)) belongs to R. i.e (x, y) ~ (u,v)

If max(x,y) = max(u,v), given that

max(x,y) = x if x >= y

= y if x < y

Prove that R is an equivalence relation

## Homework Equations

I know that to prove an equivalence relation, I must prove that the

relation satisfies : reflexivity, symmetry, and transitivity, but I

somehow get confused on how to use the condition: max(x,y) = max(u,v)

in my proof

## The Attempt at a Solution

Here are my thoughts about the proof:

To prove

Reflexivity:

Given that (x,y) ~ (u,v)

so max(x,y) = max(u,v) definitely

Symmetry:

My self-question: since (x,y) ~ (u,v). Does (u,v)~(x,y)?

My thought: since max(x,y) = max(u,v), then flip the sides

I have max(u,v) = max(x, y). This is always

true, because (x,y) ~ (u,v)

Transitivity: I get confused the most with this one, and I kind of get

stuck with the proof from here too.

My thought:

Since I'm given ((x,y), (u,v)) belongs to R and (x,y) ~ (u,v)

Can I let another ordered pairs, say (a,b) that also belongs to R

and then use the following sequence:

(x,y) ~ (u,v) and (u,v) ~ (a,b), then (x,y) ~ (a,b)?

My question: how should I introduce (a,b) into the proof? How do I

prove the fact that (u,v) ~ (a,b) is true, so that I can proceed later

steps?

The most important question: are my thoughts, so far, correct? am I

missing something?