Number of functions from one set to another?

• May 8th 2011, 08:59 AM
posix_memalign
Number of functions from one set to another?
How many functions, injections, surjections, bijections and relations from A to B are there, when A = {a, b, c}, B = {0, 1}?

Edit: I know the answer should be 64, but I don't know how to arrive at that.
• May 8th 2011, 09:13 AM
Plato
Quote:

Originally Posted by posix_memalign
How many functions, injections, surjections, bijections and relations from A to B are there, when A = {a, b, c}, B = {0, 1}?
Edit: I know the answer should be 64, but I don't know how to arrive at that.

I have no idea what you mean by 64.
There are $\displaystyle 2^3$ functions $\displaystyle A\to B.$
There are no injections $\displaystyle A\to B$.
Therefore, no bijections.
There are 6 surjections.
Because $\displaystyle \|A\times B\|=6$ there are $\displaystyle 2^6$ relations from $\displaystyle A\to B$ is you allow the empty relation,
• May 8th 2011, 09:40 AM
posix_memalign
Quote:

Originally Posted by Plato
I have no idea what you mean by 64.
There are $\displaystyle 2^3$ functions $\displaystyle A\to B.$
There are no injections $\displaystyle A\to B$.
Therefore, no bijections.
There are 6 surjections.
Because $\displaystyle \|A\times B\|=6$ there are $\displaystyle 2^6$ relations from $\displaystyle A\to B$ is you allow the empty relation,

Thanks, but why do you need the $\displaystyle \|$? Why isn't the cartesian product by itself sufficient?
• May 8th 2011, 09:46 AM
Plato
Quote:

Originally Posted by posix_memalign
Thanks, but why do you need the $\displaystyle \|$? Why isn't the cartesian product by itself sufficient?

#(X)=$\displaystyle |X|=$$\displaystyle \|X\|$ these are all common symbols standing for the number of elements in a given set.