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.