How do i prove this equivalence relation?

Question

=- is triple bar

i need to prove that [m,n] =- [j,k] iff m + k = n + j

i know that i need to have symmetry, reflexivity, and transitivity

for symmetry, would i just need to say that

m + k = n + j

m + j - (n + j) = 0

-(n + j) = -(m + j)

so n + j = m + j?

or is there something deeper that i am missing . .i also have no idea how to prove the rest .. because for reflexivity shouldn't it state that m + j = m + j, as with n + j = n + j? how can i prove this?? i don't even know where to start for transitivity .. help!!

Answer 1

Reflexivity is trivial:  since m + n = n + m for all m and n, we have [m,n] = [m,n].

Assume that [m,n] = [j,k].  Then m + k = n + j.  But then j + n = k + m, so [j,k] = [m,n], so the relation is symmetric.

Assume [m,n] = [j,k] and [j,k] = [a,b]. then m + k = n + j and

j + b = k + a.  It follows that m + k + j + b = n + j + k + a (by adding the previous equations), so that m + b = n + a, which implies that [m,n] = [a,b], so this relation is transitive.

Answer 2

Dr. Robinson wouldn't be so happy about this would he Ash? Then again, I wouldn't have found this if I weren't in a similar situation. What Tony says seems to be correct, as far as I can tell. Good luck with the rest of the homework. I absolutely despise proofs (and therefore this course lol).