Consider a binary operation # that is closed under the set of integers (if a and b are integers, then a#b is an integer).
Assume that, for all integers a and b, it is true that (a#b)#a=b.
Prove that a#(b#a)=b.
(In reply to
re(3): A simple solution by DJ)
I beg your pardon DJ.
Arguments about inverses certainly DO apply if the original problem provides the necessary basis for it.
Your "given" is equivalent to one operation "undoing" (for all integers) the other operation. Because, for any given 'a', for all integers 'b', we can break your equation down into the operation of two operators. It is very much analagous to Charlie's substitution.
In other words, the assumption GIVEN in the problem is equivalent to saying that if
F(x) = a#x, and
G(x) = x#a, then
G is the inverse of X.
Because the problem DID write G(F(x)) = x. (If we substitute the functions, we restate the problem.)
This is EXACTLY the definition of inverse. And this is EXACTLY what the problem tells us to assume.
re(4): A simple solution
