# Thread: Total relations between sets proof

1. ## Total relations between sets proof

Question: Suppose a set A has n elements and a set B has m elements. Prove that there are 2^m*n different relations from A to B

Now i see how this works with the product rule with AXB with m*n elements. and there are two sets so thats where 2 is involved. So i guess all the pieces are there but i don't know what to do with them.

2. ## Re: Total relations between sets proof

Question: Suppose a set A has n elements and a set B has m elements. Prove that there are 2^m*n different relations from A to B
I use $\|A\|$ as the notation for the number of elements is a set $A$.

Then $\|A\times B\|=\|A\|\cdot\|B\|$

Any subset of $A\times B$ is a relation $A\to B$.

So the answer to this question is $2^{\|A\|\cdot\|B\|}$

3. ## Re: Total relations between sets proof

How do you get to the point where you come to the conclusion where you use '2' to get '2^||A||*||B||' ? I'm confused on that jump.

4. ## Re: Total relations between sets proof

How do you get to the point where you come to the conclusion where you use '2' to get '2^||A||*||B||' ? I'm confused on that jump.
If S is a set, the number of subsets of S is $2^{\|S\|}$.

Here is a brief outline of a proof.
If we form the set of bit-strings, strings of 0's & 1's, of length n, then that set has $2^n$ elements.
If $S=\{a,b,c,d\}$ then the string $0,1,1,0$ represents the subset $\{b,c\}$. So every 4-bit-string represents a subset of $S$ so there are $2^4$ subsets of $S$.

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# how to find Number of relation from two sets

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