Question

How many anti symmetric relations are there on a set with \(n\) elements?

Short answer

The number should be \({2^n}3\left( {\begin{array}{*{20}{l}}n\\2\end{array}} \right)\).

Step-by-step solution

Given data

A set with \(n\) elements.

Concept used anti symmetric relation

In set theory, the relation\(R\)is said to be anti symmetric on a set\(A\), if\(xRy\)and\(yRx\)hold when\(x = y\).

Find the number of relations

The definition of an anti symmetric relation \(R\) to mean that \(aRb\) and \(bRa\) implies \(a = b\), but for a given \(a\) and \(b\) it might well be that neither \(aRb\) nor \(bRa\).

So the number should be \({2^n}3\left( {\begin{array}{*{20}{l}}n\\2\end{array}} \right)\). It is the formula for number of anti symmetric relations for \(n\) elements.