- #1
kingwinner
- 1,270
- 0
I am still struggling with the topic of cardinality, it would be nice if someone could help:
1) http://www.geocities.com/asdfasdf23135/absmath2.jpg
In the solutions , they said that{y=ax+b|a,b E R} <-> R2.
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!
1) http://www.geocities.com/asdfasdf23135/absmath2.jpg
In the solutions , they said that{y=ax+b|a,b E R} <-> R2.
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!