I'm relearning mathematics from the ground up in order to keep up with it's rigor in the university I attend. I'm doing this with serge langs basic mathematics, which thankfully involves writing proofs as well. The exercise is to let E be an abbreviation for even, and let I be an abbreviation for odd. Then you're given as a given:
E + E = E
E + I = I + E = I
I + I = E
EE = E
II = I
IE = EI = E
Then asked that you
Show that addition for E and I is associative and commutative.
Show that E plays the role if a zero element for addition. What is the additive inverse of E?
What is the additive inverse of I.
Show that multiplication for E and I is commutative and associative.
Which behaves like 0 for multiplication?
Show that multiplication is distributive with respect to addition.
I think i've answered these with something along the lines of
(E + I) + I = E + (I + I)
(I + E) + I = I + (E + I)
which I believe is technically correct but could probably be better expressed as something like
E + E + E = E + E = E = E + E = E + E + E
actually shows I associativity. you'd just commit to the same task for all different possible equations. commutativity looks like
E + E = E = E + E
what i'm confused about is how to understand this in such a case as
(EI)I = EI = E = II = I = E(II)
Which i've been led to believe is correct. However the question arises from what the equation seems to convey. How can E = II in the associated multiplicative case? E is almost like a representation of zero isnt it. I think i'm fundamentally misunderstanding something about proofs.