Let R be the relation on Z × Z, that is elements of this relation are pairs of pairs of integers, such that ((a, b),(c, d)) ∈ R if and only if a + d = b + c. Show that R is an equivalence relation.
So I know I need to show that it's reflexive, symmetric, and transitive. I have an answer but I think my wording doesn't really look proper and I think the transitive part is incorrect.
Reflexive: if a+d = b+c, then a+d = a+d, which is true.
Symmetric: if a+d = b+c then, b+c = a+d, which is true.
Transitive: if a+d = b+c and b+c = e+f, then a+d=e+f, this is true because since a+d = b+c, and b+c = e+f, then a+d must be equal to e+f
Yes, it appears you messed up on the transitive part: if $((a,b), (c,d))\in R $ and $((c,d), (e,f))\in R $, we need to show $((a,b), (e,f))\in R $. Now $a+d=b+c $ and $c+f=d+e $. So $a+f=b+c-d+d+e-c=b+e $.