Some relations behave in regular ways: "is the same age as" always holds both ways, and "is older than" passes along a chain. These patterns have names, and they are written with the quantifiers of Quantifier Order.
| Property | Formula | Example |
|---|---|---|
| Reflexive | $\forall x Rxx$ | is the same height as |
| Symmetric | $\forall x\forall y(Rxy \to Ryx)$ | is a sibling of |
| Transitive | $\forall x\forall y\forall z((Rxy \land Ryz) \to Rxz)$ | is taller than |
Key idea
Why it works
Worked examples
Worked example: Check symmetry
- Every pair must come with its reverse.
- $(b, c)$ is there, but $(c, b)$ is not.
Answer: No.
Worked example: Check transitivity
- $(a, b)$ and $(b, c)$ are in $R$, so $(a, c)$ must be.
- It is not.
Answer: No. (Also $(a, b)$ and $(b, a)$ would require $(a, a)$.)
Worked example: Everyday relations
- Not reflexive: no one is their own sibling.
- Symmetric: if Ava is Ben's sibling, Ben is Ava's.
- Not transitive: Ava is Ben's sibling and Ben is Ava's, but Ava is not her own sibling.
Answer: Symmetric only.
Common mistake
Mathematics is full of these properties. "Equals" is reflexive, symmetric and transitive (a relation with all three is called an equivalence relation). "Is less than" is transitive but neither reflexive nor symmetric. "Divides", on whole numbers, is reflexive and transitive. Checking the three properties is often the first thing a mathematician does with a new relation.
- To refute a property, find one pair (or triple) that breaks it.
- To confirm it, check every pair (or every chain of two arrows).
- Loops ($x$ to $x$) matter for reflexivity and can matter for transitivity.