Relation and the complementary relation: reflexivity and irreflexivity

Question

How do I show that the relation R on a set A is reflexive if and only if the complementary relation R is irreflexive.

Because of iff:

1. I start with let R be a relation from a set A to B. The complementary relation R is the set $\{(a,b):(a,b) \not\in R\}$

2. I think a relation R on the set A is irreflexive if for every a is an element of A, $(a,a)$ is in R. That is, R is irreflexive if no element in A is related to itself.

Answer 1

It’s really just a matter of looking at the definitions.

• $R$ is reflexive if $\langle a,a\rangle\in R$ for each $a\in A$.
• $R$ is irreflexive if $\langle a,a\rangle\notin R$ for each $a\in A$.

I’ll write $R^c$ for the complementary relation, and I’ll do half of it.

$$R^c=(A\times A)\setminus R=\{\langle a,b\rangle\in A\times A:\langle a,b\rangle\notin R\}\;,$$

so if $\langle a,a\rangle\in R$, then $\langle a,a\rangle\notin R^c$ and vice versa. Thus, if $\langle a,a\rangle\in R$ for every $a\in A$, then $\langle a,a\rangle\notin R^c$ for each $a\in A$, and therefore by definition $R^c$ is irreflexive.

Now you just have to explain why $R$ is reflexive if $R^c$ is irreflexive. All of the pieces are there; you just have to put them together properly.