Thread: [SOLVED] A few intro to advanced math topics

Ok, so I have a few problems that I need to work on and have some progress made by Monday. The problem is that I need to work 30+ hours this weekend. So thanks in advance to anyone who assissts in solving, offers some advice, or views this topic

1) Let f: X--> Y and let A,B be subsets of Y. Prove that f^-1(A) - f^-1(B)= f^-1 (A-B).

Kind of a fun problem. Just show containment both ways.

2) Let f: X-->A and g: Y-->B. Define a function f x g by (f x g)(x,y)= (f(x),g(y)) for all (x,y) as elements in X x Y.

note that f x g is f cross g, not f times g.

Prove the following
a) If f and g are one-ton-one, then f x g is one-to-one.

b) If f maps onto A and g maps onto B then f x g maps on A x B

3) True or False? If false give a specific counter example (So indicate exactly what the sets and functions involved are) Assume f: X --> Y and
g: Y--> Z

a) If F^-1(Y) = X, then f maps onto Y

b) If g composed with f maps onto Z, then f maps onto Y

Again, thank you very much

f^-1(A) - f^-1(B)= f^-1 (A-B).

For :

Let .
Then
and ,
that is
and
so

or
.

Try the other inclusion the same way.

If f and g are one-ton-one, then f x g is one-to-one.

Just let
then

or
and

Since now f,g are both one to one, we get....

If f maps onto A and g maps onto B then f x g maps on A x B

Let . We wish to find such that .

Now since and maps onto , there exists some such that .

Similarly, there must be some such that .

Now, check what is.


Ok, this is where latex officialy bores me

If F^-1(Y) = X, then f maps onto Y

Let X={1} and Y={2,3}. Define f(1)=2; This f is not onto. But the inverse image of Y={2,3} through f is...

If g composed with f maps onto Z, then f maps onto Y

With X,Y as before, define now Z={4} and g(2)=g(3)=4. Then gof (composition) maps X onto Z, but...