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CHAPTER 11
Relations
In mathematics there are endless ways that two entities can be related
to each other. Consider the following mathematical statements.
5 < 10 5 ≤ 5 6 = 30 5
5 | 80 7 > 4 x ≠ y 8 ∤ 3
a ≡ b ( mod n) 6 ∈ Z X ⊆ Y π ≈ 3.14 0 ≥ −1 2 ∉ Z Z ⊈ N
In each case two entities appear on either side of a symbol, and we
interpret the symbol as expressing some relationship between the two
entities. Symbols such as , ∈ and ⊂, etc., are called relations
because they convey relationships among things.
Relations are significant. In fact, you would have to admit that there
would be precious little left of mathematics if we took away all the relations.
Therefore it is important to have a firm understanding of relations, and
this chapter is intended to develop that understanding.
Rather than focusing on each relation individually (an impossible task
anyway since there are infinitely many different relations), we will develop
a general theory that encompasses all relations. Understanding this
general theory will give us the conceptual framework and language needed
to understand and discuss any specific relation.
Before stating the theoretical definition of a relation, let’s look at a
motivational example. This example will lead naturally to our definition.
Consider the set A = { 1,2,3,4,5 } . (There’s nothing special about this
particular set; any set of numbers would do for this example.) Elements of
A can be compared to each other by the symbol “