I have a final tonight and I forgot how to do this:
Let f: X Y be a function. Show that
(a) f(S T)) f(S) f(T)
(b) If f is 1-1, then show that the above inclusion is actually an equality of sets.
(a) f(S T) { y | y = f(x), x S and x T} {y | y = f(x), x S} and {y | y = f(x), x T} f(S) f(T)
(b) Is the following correct?
If f 1-1 then f( ) = f( ) =
We have already shown that f(S T)) f(S) f(T). It is left to show that f(S) f(T) f(S T)).
If f(x) f(S) f(T), then f(x) f(S) and f(x) f(T). Because f is 1-1, x S and x T. This means x S T. Again because f is 1-1, f(x) f(S T). This proves each is a subset of the other so they are equivalent.