Although we have tried to regard relations as mathematical objects in their own right, they have been somewhat tenuous objects, plucked from our intuitions of prop-erties. Most people find a degree of tension in the use of the word “thing” in the phrase “a relation is a thing that links objects together.”
In mathematics, it is common to objectify abstractions of this sort by means of a formal mathematical definition of a model. The model that is usually taken for a relation is as a set of ordered pairs. IfR is a relation between the two sets A and B, we modelR by means of the set:
ˆR = {(a, b) ∈ A × B | aRb}
Thus, for any elementsa∈ A, b ∈ B,
aRb iff (a, b)∈ ˆR
Thus, the question as to whether or notaRb for a given pair a, b can be replaced by the set-membership question as to whether or not(a, b)∈ ˆR.
Now there are a great many questions one can ask about relations. Is it a relation used by people, and if so what kind of people use it? Is it a legal relation, and if so under what legal system? Is it a purely mathematical relation, and if so who invented it? And so on. But for most mathematical applications, none of these questions is important. In mathematics, the only question of significance as far as relations are concerned is: which pairs of elements satisfy the relation and which do not? But this is precisely the question that can be answered in straight set-theoretic terms by looking at the model set ˆR.
In other words, as far as mathematics is concerned, all that you need is the set ˆR of those pairs of objects that are linked by the relationR. In other words, you don’t need (real) relations at all, it is enough to have the corresponding sets.
Accordingly, it is common in mathematics to “forget” that relations are relations, and to think of them instead as sets of ordered pairs. That is to say, we refer to the model sets ˆR as “relations” and forget about R altogether. This gives us the following alternative definition:
A relation between a setA and a set B is a subset of the Cartesian product A× B.
In particular, a relation on a setA is a subset of A× A.
Having made the switch from real relations to their set-theoretic counterparts, we now have no further need for the notation ˆR, of course. We can just speak about “the relationR,” it now being understood that R will be a set of ordered pairs.
In terms of our new definition, the properties of reflexivity, symmetry, and transi-tivity for a relationR on a set A now read as follows:
• R is reflexive iff
(∀a ∈ A)[(a, a) ∈ R]
• R is symmetric iff
(∀a, b ∈ A)[(a, b) ∈ R → (b, a) ∈ R]
• R is transitive iff
(∀a, b, c ∈ A)[((a, b) ∈ R ∧ (b, c) ∈ R) → ((a, c) ∈ R)]
At least, this is the new official way of writing these definitions. In practice, we continue to use our previous, less cumbersome infix notation, the only difference being that this is now just an abbreviation. That is to say, we agree to writeaRb in place of(a, b)∈ R.
This takes us back to our starting point, except that now we no longer have to accept relations as some kind of strange objects; they are just sets — specifically, subsets of Cartesian products.
In fact we now have far more freedom than before in constructing relations. Ac-cording to our new definition, any subset ofA× B will be a relation between A and B, regardless of whether or not there is some nice way to describe this relation.
For example, we may define a relation by simply listing the ordered pairs it con-tains, such as the relationR between the sets{1, 3, 5, 7, 9} and {2, 4, 6, 8} defined by:
R= {(1, 2), (3, 2), (9, 2), (3, 6), (3, 8), (7, 2), (7, 4), (7, 6), (7, 8)}
This issue is taken up again in the exercises for this section.
It should be noted that the above procedure of replacing a rather abstract notion by a formal mathematical model is a common one in mathematics. In fact we already met this device in Chapter 2, when we replaced the complex notion of implication by the formally defined (material) conditional⇒. Just as the conditional captures something, but not everything, about implication, so too our definition of a relation as a set of ordered pairs captures something, but not everything, about real relations.
Thus, our (new) definition of, say, the relation of marriage as being simply the set {(a, b) | a is married to b}
certainly does not capture everything about the relationship of marriage; but it does capture enough for our present, mathematical purposes.
Exercises 5.3
From now on, we always assume that a relation is a set of ordered pairs.
(1) LetA= {1, 2, 3}. Define (by explicitly writing down the set of ordered pairs) a relation onA that is:
(a) reflexive, not symmetric, not transitive;
(b) not reflexive, symmetric, not transitive;
(c) not reflexive, not symmetric, transitive;
(d) not reflexive, symmetric, transitive;
(e) reflexive, not symmetric, transitive;
(f) reflexive, symmetric, not transitive;
(g) not reflexive, not symmetric, not transitive;
(h) reflexive, symmetric, transitive.
(2) Which of the properties reflexive, symmetric, transitive do the following rela-tions satisfy?
(a) R onR × R defined by
(x, y)R(z, w) iff x+ z ≤ y + w (b) R onR × R defined by
(x, y)R(z, w) iff x+ y ≤ z + w
(3) LetR be a relation on A such that (∀a ∈ A)(∃b ∈ A)[aRb]. Show that if R is symmetric and transitive, thenR is reflexive.
(4) Is it true that ifR and S are relations on A× B, then R ∪ S and R ∩ S are relations onA× B? Justify your answer.
(5) Which of the following, if any, are true? (Justify your answer in each case.) (a) IfR, S are reflexive relations on A, then R∪ S is a reflexive relation
onA.
(b) IfR, S are reflexive relations on A, then R∩ S is a reflexive relation onA.
(c) IfR, S are symmetric relations on A, then R∪ S is a symmetric relation onA.
(d) IfR, S are symmetric relations on A, then R∩ S is a symmetric relation onA.
(e) IfR, S are transitive relations on A, then R∪ S is a transitive relation onA.
(f) IfR, S are transitive relations on A, then R∩ S is a transitive relation onA.
(6) A relationR on a set A is said to be asymmetric iff (∀a, b ∈ A)[aRb → ¬(bRa)]
and is said to be antisymmetric if
(∀a, b ∈ A)[(aRb ∧ bRa) → (a = b)]
Prove that ifR is asymmetric, then it is antisymmetric. What can you say about the converse?
(7) A relation onA that is reflexive, antisymmetric, and transitive is called a par-tial order onA. Prove that each of the ordering relations < and≤ are partial orders onR and that ⊂ and ⊆ are partial orderings on P(X) for any set X.