1. The problem statement, all variables and given/known data 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 2. Relevant 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 3. 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?