Existence Statements - Mathematical Sentences

Mathematical Writing - Vivaldi Franco 2014

Existence Statements
Mathematical Sentences

An existence statement is a logical expression with a leading existential quantifier $$\exists $$. In this section we consider some examples; we shall take a deeper look at the question of existence in Chap. 9.

The standard symbolic form of an existence statement is (4.18), but in a mathematical sentence the quantifier is often hidden:

The number $$\cos (\pi /3)$$ is rational.

To make the quantifier visible, we write

For some rational number $$r$$, we have $$\cos (\pi /3)=r$$.

or, in symbols,

$$ \exists r\in \mathbb {Q},\,\,r=\cos (\pi /3). $$

This statement is weaker than the identity $$\cos (\pi /3)=1/2$$, because it does not require us to reveal which rational number our expression is equal to.

Equation (4.20) shows that an existential quantifier is hidden in the definition of the divisibility of integers. Therefore the sentence ’$$n$$ is even’, or, more generally, ’$$m$$ divides $$n$$’ are existence statements.

The sentence ’$$n$$ is not prime’ is an existence statement, because it means ’there is a proper divisor of $$n$$’. We write it symbolically as

$$\begin{aligned}&\exists m\in \mathbb {N},\,\,m\not \in \{1,n\} \wedge (m | n)\\&\qquad \Leftrightarrow \,\,\exists m\in \mathbb {N},\,\,m\not \in \{1,n\} \wedge (\exists k\in \mathbb {N},\,\,mk=n) \end{aligned}$$

where we have been careful in excluding trivial divisors.

The following well-known theorem in arithmetic is an existence statement.

Theorem

(Lagrange,4 1770). Every natural number is the sum of four squares.

Let us analyse it in detail. First, we consider two instances of the theorem:

$$ 5=2^2+1^2+0^2+0^2\quad \qquad 7=2^2+1^2+1^2+1^2. $$

The first identity shows that some integers are the sum of fewer than four squares, while one checks that 7 requires all four squares. We restate the second identity without disclosing the details:

$$ \exists \, a,b,c,d\in \mathbb {Z},\,\,7=a^2+b^2+c^2+d^2. $$

By replacing 5 or 7 with an unspecified natural number $$n$$, we obtain a predicate $${\fancyscript{L}}$$ over $$\mathbb {N}$$:

$$\begin{aligned} {\fancyscript{L}}(n):=\left( \exists \, a,b,c,d\in \mathbb {Z},\,\,n=a^2+b^2+c^2+d^2 \right) \end{aligned}$$

(4.24)

or

$$n$$ is a sum of four squares.

To state Lagrange’s theorem with symbols, we quantify the remaining variable $$n$$:

$$\begin{aligned} \forall n\in \mathbb {N},\,\,\exists a,b,c,d\in \mathbb {Z}, \,\,n=a^2+b^2+c^2+d^2. \end{aligned}$$

(4.25)

Five quantifiers are necessary to turn the predicate $$n=a^2+b^2+c^2+d^2$$ (a function of five variables) into a boolean constant. In the original formulation of this theorem, all five variables have fallen silent.

Having buried all meaning inside the predicate (4.25), Lagrange’s theorem now disappears into a set identity:

$$ \mathbb {N}=\{n\in \mathbb {N}\,:\,{\fancyscript{L}}(n)\}. $$

Any theorem on natural numbers can be put in this form for an appropriate predicate $${\fancyscript{L}}$$. (Think about it.)

Negating expressions with a leading universal quantifier always results in existence statements, as we shall see in the next section.