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1) http://www.geocities.com/asdfasdf23135/absmath2.jpg

In the solutions , they said that{y=ax+b|a,b E R} <-> R

^{2}.

But I am wondering...what is the actual mapping that gives one-to-one correspondence (1-1, onto)? How can we define such a map?

Also, I don't understand the last line of the solutions at all. In particular, c+c=c? How come? I am feeling very uncomfortable about this...

**2) Show that if S and T are finite sets, then the set of all functions from S to T has |T|**

^{|S|}many elements.__My idea:__Fix an element of S, there are |T| possible ways to map this element of S to an element of T. S has |S| many elements, so we have |T| x |T| x ... x |T| (|S| times)=|T|

^{|S|}ways to determine a function from S to T, and thus |T|

^{|S|}elements in the set of all functions from S to T.

But what if S and T are "empty sets"? How can I prove the statement in these cases?

Thanks for any help!