2.3 Quantifiers

By way of motivation, consider the following statement¹¹ 1 Logicians might not like us calling $A$ a statement at all because it involves $n$ as a ‘free variable’. You do not need to be worried about this at this stage of your mathematical career..

$A$ : ‘The square of the integer $n$ is even.’

We label this statement $A$ so we can refer to it concisely. Whether $A$ is true or false depends on the value of $n$ . For example,

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if $n=4$ then $A$ is true (because $4^{2}=16$ , which is even);
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if $n=5$ then $A$ is false (because $5^{2}=25$ , which is not even).

We cannot know whether $A$ is true without knowing something about the value of $n$ .

In statement $A$ , the symbol ‘ $n$ ’ is called a variable because its value can vary. Symbols can also be used to stand for constants, e.g. $\pi$ and $e$ . The value of a constant is fixed.

As mathematicians, we aim to make statements and establish that they are true (by giving a proof) in order to advance our understanding of the subject. Statement $A$ does not really help our understanding of which integers have even squares because it is sometimes true and sometimes false. Compare it with the following two statements:

$B$ : ‘Let $n$ be $2$ , $4$ or $6$ . The square of the integer $n$ is even.’ [True]

$C$ : ‘Let $n$ be $1$ , $2$ or $3$ . The square of the integer $n$ is even.’ [False]

We can say for sure that statements $B$ and $C$ are either true or false, but not both, because we have specified something about the variable $n$ . We say that the variable has been quantified.

Two types of quantifier are very common in mathematics. These are ‘for all’ (the universal quantifier) and ‘there exists’ (the existential quantifier). You have probably seen many statements that use these quantifiers already in your mathematical career, but you might not have thought about them in such terms.

2.3.1 Universal quantifier

Consider the following claim.

Claim 2.8.

For all positive real numbers $x$ , $x^{2}+x$ is positive.

The phrase ‘for all’ is called the universal quantifier. It is used when we want to make a statement about all objects with some property – in this case all positive real numbers. There are different ways of phrasing Claim 2.8, for example:

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Given any positive real number $x$ , $x^{2}+x$ is positive.
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Suppose $x$ is a real number. If $x>0$ then $x^{2}+x>0$ .

Read these carefully and convince yourself that they mean the same thing as Claim 2.8.

The point here is that there is room for mathematicians to have their own writing style while working within the norms expected by the rest of the mathematical community. You will find a style that suits you as you progress through your degree.

Some people use $\forall$ (an upside-down A) as a shorthand for the phrase ‘for all’.

Here are two more examples of statements using a universal quantifier:

Claim 2.9.

For all real numbers $x$ , $x^{2}+x$ is positive.

Claim 2.10.

For all real numbers $x$ strictly between $-1$ and $0$ , $x^{2}+x$ is negative.

Exercise 2.11.

Decide whether you think Claims 2.8, 2.9 and 2.10 are true or false.

Try rephrasing the claims in different, but equivalent, ways. Which way seems most natural to you?

2.3.2 Existential quantifier

Consider the following claim.

Claim 2.12.

There exists a real number $x$ such that $x^{2}+x$ is positive.

The phrase ‘there exists’ is called the existential quantifier. It is used when we want to make a statement about there being at least one object with some property. There are different ways of phrasing Claim 2.12, for example:

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There is a real number $x$ with the property that $x^{2}+x$ is positive.
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Some real number $x$ is such that $x^{2}+x>0$ .

Read these carefully and convince yourself that they mean the same thing as Claim 2.12.

Some people use $\exists$ (a back-to-front E) as a shorthand for the phrase ‘there exists’.

Note that Claim 2.12 does not state precisely which real numbers make $x^{2}+x$ positive, or exactly how many there are; it just says that there is at least one value of $x$ which does.

Here are two more examples of statements using an existential quantifier:

Claim 2.13.

There is a real number $x$ such that $x^{2}+x$ is negative.

Claim 2.14.

For some real number $x$ strictly between $-1$ and $0$ , $x^{2}+x$ is positive.

Exercise 2.15.

Decide whether you think Claims 2.12, 2.13 and 2.14 are true or false.

Try rephrasing the claims in different, but equivalent, ways. Which way seems most natural to you?