• Let S be a set and ~ a relation between any two objects in the set:
• Reflexivity For every a in S, a ~ a
• Symmetry For any a, b in S, if a ~ b -> b ~ a
• Transitivity For any a, b, c in S, if a ~ b and b ~ c –> a ~ c

Reflexivity: everything is related to itself

Reflexivity is the smallest of the three rules, and the easiest to wave past. Which is exactly why I want to slow down on it. A relation is reflexive when nothing is a stranger to itself.

The core idea, in plain words

A relation $\sim$ on a set $S$ is just a rule that says, for each ordered pair $(a, b)$, whether $a \sim b$ holds. It is reflexive when

Every single element must be related to itself. Not most of them. Every one. A single element that isn’t spoils it.

An everyday analogy: “is the same height as”

Am I the same height as myself? Obviously. So is everyone else. “Is the same height as” is reflexive. Now try “is taller than”. Am I taller than myself? No. Nobody is. That relation is not merely not reflexive; it is irreflexive: no element is related to itself.

A small worked example

Let $S = \{1, 2, 3\}$ and write a relation as a set of ordered pairs.

• $R_1 = \{(1,1), (2,2), (3,3), (1,2)\}$: reflexive ✔ (all three “self-loops” are there; the extra pair $(1,2)$ does no harm).
• $R_2 = \{(1,1), (2,2), (1,2)\}$: not reflexive ✘, because $(3,3)$ is missing.
• $R_3 = \{(1,2), (2,3)\}$: irreflexive (no self-loops at all).
• $R_2$ is also a useful warning: it is neither reflexive nor irreflexive. These two words are not opposites.

Some familiar ones on the integers:

A picture in words: loops on every dot

Draw each element as a dot and each related pair $(a,b)$ as an arrow from $a$ to $b$. Reflexive means every dot has a little loop back to itself. In the table (matrix) picture, it means the whole main diagonal is filled in.

graph LR
  1((1)) --> 1
  2((2)) --> 2
  3((3)) --> 3
  1 --> 2

Common misconceptions

• “Reflexive just means $a \sim a$ is possible.” No: it must hold for every $a$ in the set. And the set matters: “$a \le b$” is reflexive on any set of numbers, but “$a$ has the same parents as $b$” is only reflexive on a set where everyone has known parents. Always ask: on which set?
• “Not reflexive means irreflexive.” No: $R_2$ above is neither.
• “Symmetric + transitive gives reflexive for free.” A famous false proof: “from $a \sim b$ get $b \sim a$ by symmetry, then $a \sim a$ by transitivity.” The flaw: it assumes every $a$ is related to something. The empty relation on $\{1\}$ is symmetric and transitive (vacuously) but not reflexive. See Symmetry.

How it connects

• Reflexivity is one of the three legs of an equivalence relation (with Symmetry and Transitivity) and one of the three legs of a partial order (with antisymmetry and transitivity). Both live on Set Relations.
• In Groups, “$a$ and $b$ are in the same coset of $H$” is reflexive precisely because the identity is in $H$.

The truth revealed: reflexivity is the promise that everything counts as itself. It sounds trivial until you meet a relation that breaks it.