Question

How to use a zero-one matrix to represent a relation to determine whether the relation is reflexive, symmetric or antisymmetric?

Short answer

Answer is detailed below.

Step-by-step solution

Given data

A zero-one matrix to represent a relation.

Concept used relation

A relation between two sets is a collection of ordered pairs containing one object from each set.

Define relation

A reflexive relation must have all ones on the main diagonal, because we need to have \((a,a)\) in the relation for every element \(a\).

A symmetric relation must have the same entries above and below the diagonal, that is, a symmetric matrix remains the same if we switch rows with columns. This is because by definition, if \((a,b)\) ? R, then \((b,a)\) must also be in \(R\).

In an anti symmetric matrix, if there is a 1 above or below the main diagonal, there must be a zero as the mirror image on the other side of the diagonal. By definition, a relation \({\rm{R}}\) is anti symmetric if and only if we have: \((a,b) \in R \wedge (b,a) \in R \Rightarrow a = b\)