Assumptions are the termites of relationships.

character of Arthur Fonzarelli, *Happy Days*

Now we are ready to consider some properties of relations. Our interest is to find properties of, e.g. *motherhood*. To do this, remember that we are not interested in a particular mother or a particular child, or even in a particular mother-child pair, but rather motherhood in general.

We'll start with properties that make sense for relations whose source and target are the same, that is, relations on a set.

**Definition. **Let be a relation on the set .

- We call
*reflexive*if every element of is related to itself; that is, if every has . - We call
*irreflexive*if no element of is related to itself. - We call
*symmetric*if means the same thing as . - We call
*asymmetric*if guarantees that . - We call
*antisymmetric*if the only way for and to both be true is if . - We call
*transitive*if and together guarantee .

**Exercise. **Explain why none of these relations makes sense unless the source and target of are the same set.

**Exercise. **Which of the above properties does the motherhood relation have?

**Exercise.** Write the definitions above using set notation instead of infix notation.

**Exercise. **Write the definitions of reflexive, symmetric, and transitive using logical symbols.

Checking whether a given relation has the properties above looks like:

**E.g. **`Divides' (as a relation on the integers) is reflexive and transitive, but none of: symmetric, asymmetric, antisymmetric.

**Proof. **We'll show reflexivity first. Suppose is an integer. Then , so divides .

Now we'll show transitivity. Suppose divides and divides . Then there are and so that and . Then , so divides .

Note that 2 divides 4 but 4 does not divide 2. This counterexample shows that `divides' is not symmetric.

Note that 4 divides 4. This counterexample shows that `divides' is not asymmetric.

Note that divides and divides , but . This counterexample shows that `divides' is not antisymmetric.

**E.g. ** is irreflexive, asymmetric, transitive, and antisymmetric, but neither reflexive nor symmetric.

**Exercise. **Show that `divides' *as a relation on * is antisymmetric.