# Isn't reflexivity and symmetry implied in equivalence relations?

It looks like for all "nice" sets, the set $S\times S$ will have symmetry and reflexivity by default. The tough part is usually showing transitivity. However, are there any non-empty sets such that $S\times S$ is not symmetric and reflexive?

• What exactly is the equivalence relation you're working with here? – Alfred Yerger May 7 '15 at 4:31
• The set $S\times S$ is always an equivalence relation, and is reflexive, symmetric, and transitive: it is the relation that says that every element of $S$ is related to every element of $S$. Perhaps you meant something else? Some subset of $S\times S$ perhaps? – MJD May 7 '15 at 4:31
• If you take $S \times S$ to be the (equivalence) relation then it will always be trivially reflexive, symmetric and transitive for any set $S$. It is only when you take a non-trivial relation that this might not happen. – Rogelio Molina May 7 '15 at 4:32
• @MJD I am afraid you answered my question. – SalmonKiller May 7 '15 at 4:32
• I'm glad I could help! – MJD May 7 '15 at 4:32

The set $S\times S$ is always an equivalence relation, and is reflexive, symmetric, and transitive: it is the relation that says that every element of $S$ is related to every element of $S$.

To answer your question from the comments, suppose we have some set $S$, and we have a relation on $S$, say $\sim$. We want to represent $\sim$ as a subset $R_\sim\subset S\times S$, and your question is which subset $R_\sim$ corresponds to the relation $\sim$. The correspondence is very straightforward: We have $a \sim b$ if and only if $\langle a,b\rangle \in R_\sim$.

For example, let $R_\le$ be the subset of $\Bbb Z\times \Bbb Z$ that represents the $\le$ relation. Then $\langle 1,2\rangle$ and $\langle 1,17\rangle$ and $\langle 9,9\rangle$ are elements of $R_\le$, because $1\le2$ and $1\le 17$ and $9\le 9$, but $\langle 3,1\rangle$ is not because $3\not\le 1$.

When $R = S\times S$, we have that $\langle a,b\rangle\in R$ for all $a$ and $b$, and so the relation represented by this $R$ has $a\sim b$ for all $a$ and $b$—that is, $a$ and $b$ are always related. It should be easy to prove that this relation is reflexive, symmetric, and transitive. Similarly, when $R = \emptyset$, we get the relation where $a$ and $b$ are never related, which is symmetric and transitive, but not reflexive.

• So when I see $b \in X, a \sim b$, I should be thinking what? Because I understand that $\sim$ is a set, but what does it mean for $a$ and $b$ to be related in that way? – SalmonKiller May 7 '15 at 4:35
• @SalmonKiller That needs some context. The point here is that certain subsets of $S \times S$ may fail reflexivity and symmetry. – Dustan Levenstein May 7 '15 at 4:37
• @DustanLevenstein The context is equivalence classes. They are defined as $\lbrace x \in X; a\sim x \rbrace$ – SalmonKiller May 7 '15 at 4:38
• That's the definition of the equivalence class containing $a$ (assuming an equivalence relation on $X$). The equivalence classes partition $X$, meaning that $X$ is the disjoint union of those equivalence classes. – Dustan Levenstein May 7 '15 at 4:41
• Can you clarify what it means for a Cartesian product to be an equivalence relation? I don't understand. Edit: Now I see it. Thanks. I've never seen this notation before! – Alfred Yerger May 7 '15 at 4:42