Number of functions from one set to another?

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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 $2^3$ functions $A\to B.$
There are no injections $A\to B$ .
Therefore, no bijections.
There are 6 surjections.
Because $\|A\times B\|=6$ there are $2^6$ relations from $A\to B$ 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 $2^3$ functions $A\to B.$
There are no injections $A\to B$ .
Therefore, no bijections.
There are 6 surjections.
Because $\|A\times B\|=6$ there are $2^6$ relations from $A\to B$ is you allow the empty relation,

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

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