What is the Indexed Family of Subsets?

[pezola]

Homework Statement

Let f: A \rightarrow B be given and let {X_{\alpha}} for \alpha \in I be an indexed family of subsets of A.


Prove:

a) f(U_{\alpha\inI} X_{\alpha}) = U_{\alpha\inI}f(X_{\alpha})

The Attempt at a Solution

To prove these two things are equal I must show that the left side is a subset of the right and that the right side is a subset of the left. However, the notation on these problems is really confusing me. I understand that I am being asked to show that the function f applied to the union of all the X_{\alpha} s is equal to the union of what you get after you apply the function f to the X_{\alpha}s. And the result seems reasonable to me, but I have no idea how to right this out.

There are actually many more parts to this question, but I think I will be able to do them once I understand how to write things out.

(I had a hard time getting the symbols to type in right, so I am including a scanned version of the problem as well. Thanks.)

 

[tiny-tim]

Let f: A \rightarrow B be given and let {X_{\alpha}} for \alpha \in I be an indexed family of subsets of A.

Prove:

a) f(U_{\alpha\inI} X_{\alpha}) = U_{\alpha\inI}f(X_{\alpha})

Hi pezola!

(you have to leave a space after \in )

Hint: start "if y ∈ f(\bigcup_{\alpha\in I}X_{\alpha}), then ∃ x ∈ \bigcup_{\alpha\in I}X_{\alpha} such that f(x) = y, so ∃ … "

 

[pezola]

It is amazing the difference a space will make.

Thank you so much!...I was able to do all parts of the problem and even presented them in class!