# Why are both of these not equivalence relations?

Can anyone tell me why the first set is an equivalence relation, and not the second? As far as I can see, both are reflexive, symmetric and transitive, but my books says only the second one is an equivalence relation. The set is {1,2,3,4}

{(1,4),(4,1),(2,2),(4,4)}
{(1,1),(2,2),(2,3),(3,2),(3,3),(4,4)}

## 2 Answers

Since the underlying set is $\{1,2,3,4\}$, the first relation is not reflexive: it’s missing the pairs $\langle 1,1\rangle$ and $\langle 3,3\rangle$. It’s also not transitive, since it has $\langle 1,4\rangle$ and $\langle 4,1\rangle$ but not $\langle 1,1\rangle$.

• I see, but is it required for an equivalence relation to include all the given elements in the set? – user103548 Nov 4 '13 at 10:31
• @user103548: A relation on $A$ is reflexive if and only if it includes $\langle a,a\rangle$ for every $a\in A$. – Brian M. Scott Nov 4 '13 at 10:32

The first is not an equivalence relation because it is not reflexive. It is missing $(1,1),(3,3)$. The lack of $(1,1)$ also implies that it is not transitive, because we have both $(1,4),(4,1)$. The second set, however, is an equivalence relation.

For reference, here's a definition:

A relation $R$ on a set $X$ is a subset of $X \times X$.

An equivalence relation on $X$ is a relation $R$ on $X$ such that:

• Reflexivity: For all $a \in X, (a,a) \in R$
• Symmetry: For all $a,b \in X$, if $(a,b) \in R$, then $(b,a) \in R$.
• Transitivity: For all $a,b,c\in X$, if $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$.