An Introduction to Mathematical Reasoning Numbers, Sets and Functions

In Chapter 1 a predicate was described as an expression containing one or more free variables; it becomes a proposition, and so is true or false, when a specific value is assigned to each free variable. Of course whether this proposition is true or is false usually depends on the values selected.

However, we saw in the last chapter that a proposition can be created from a predicate in another way – by making a statement about the set of values of the free variables which make it true. Many results in mathematics take the form of listing the values of this set, for example when we solve an equation. But often results simply address the question of whether there is any choice of values of the free variables resulting in a true proposition and whether there is any choice resulting in a false proposition. Statements that such values exist are known as existential statements. Statements that they do not can be thought of as universal statements. We met examples of universal statements when discussing implications in Chapter 2.

In this chapter we discuss general universal and existential statements.

Universal statements

Suppose that P(a) is a predicate with a single free variable a with possible values in a set A. Usually P(a) is true for some elements of the set A and false for others, and in the last chapter we described how such a predicate could be used to describe a subset of A, denoted by {a ∈ A ∣ P(a)}, the subset of elements of A for which the statement P(a) is true.