# Antisimmetric relations question

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1. Apr 9, 2017

### doktorwho

1. The problem statement, all variables and given/known data
A set $P=\left\{ \ p1,p2,p3,p4 \right\}$ is given. Determine the number of antisimmetric relations of this set so that $p1$ is in relation with $p3$, $p2$ is in relation with $p4$ but $p2$ is not in relation with $p1$.
2. Relevant equations
3. The attempt at a solution

By drawing the table of relations i concluded that the total number of antisimmetrical relations without any constriction further implied in the problem is $2^n3^{\frac{n^2-n}{2}}$ with the first factor drawn from the diagonal possibilities and the second factpr being 3 possibilities on the upper part above the diagonal. I do not however know how to include these steps. Can you help?

2. Apr 9, 2017

### nuuskur

This is the picture as I understand it.
$$\begin{array}{ccc}x&x&1&x\\0&x&x&1\\x&x&x&x\\x&x&x&x\end{array}$$
You correctly deduce that there are a total of $2^n\cdot 3^{\binom{n}{2}}$ anti-symmetrical relations on a set of $n$ elements. There's nothing special to this, you have fixed 2 components in the upper triangle bit. Keep in mind what an anti-symmetrical relation matrix has to look like and use the product rule.