S is a set and * is a binary operation on S, with identity e in S. Also, for all x, y, z elements of S, x*(y*z)=(x*z)*y. Prove * is commutative.
my first idea was to let z=e, but that only proves it for that specific case....I feel like this is relatively simple compared to my other proofs, I'm just missing something on this one :( thanks in advance for your help!
You need to remember what a binary operator does, hence it takes two elements from S and it returns a single element.
Notice that the probelm provides two inputs, an associative binary operation xo(yoz)=(xoy)oz and an identity element, hence the set that posssesses these features is a monoid.
You may test if the monoid is commutative using the following binary operation such that:
`x o y = xy + x + y`
If x and y are positive integers, using the binary operation yields:
`xoy = xy + x +`` y`
`y o x = yx +y + x`
Since addition and multiplication are commutative over the positive set of integers, then the binary operation is commutative.
Hence, using the features provided by problem yields that the binary operation `o` is also commutative over the set S.