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I am trying to learn the basics of combinatorics and having a bit of difficulty grasping the concepts. I am having some trouble understanding the following problem:

$A$ and $B$ are both finite sets.

For elements $x$ and $y$, how many functions $f: A\to B$ exist such that

$f(x) = f(y)$ where $x,y$ $\epsilon$ $A$ ?

What I have attempted is the following:

Let $N$ be the set of all functions that satisfy the above property and $U$ be the total number of functions between $A$ and $B$. Then $|N| = |U| - |U/N|$. I feel uncertain as to whether my logic is correct in this solution.

How might the logical approach to this problem change if the property was $f(x) \ne f(y)$ instead?

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    It's the similar to functions from A to B with only condition that f(x)=f(y) so one less possibility $|B|^{|A|-1}$ – 2017-01-21

1 Answers 1

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there are $|B|^{|A|-1}$ such functions, since each one is uniquely determined by the images of the elements in the set $A\setminus\{x\}$ and these can be picked freely.