Relational Operators - Mathematical Sentences

Mathematical Writing - Vivaldi Franco 2014

Relational Operators
Mathematical Sentences

Franco Vivaldi1

(1)

School of Mathematical Sciences, Queen Mary, University of London, London, UK

Franco Vivaldi

Email: f.vivaldi@qmul.ac.uk

Consider the following theorem of analysis:

$$\begin{aligned} \textit{Let}\ \ f:\mathbb {R}\rightarrow \mathbb {R}\;\textit{be a differentiable function. Then f is continuous.} \end{aligned}$$

(4.1)

This statement comprises two sentences. The first sentence does not state a fact: it’s an assumption. We aren’t told which function $$f$$ is, so there is no question of this statement about $$f$$ being true or false. But still we use it as a basis for the rest of the argument. The second sentence does just that: it’s a deduction of a new fact from the assumption. The symbol $$f$$ is an internal variable of the statement, and we can make it disappear:

Every differentiable real function is continuous.

This statement is equivalent to (4.1).

In this chapter we begin to analyse mathematical sentences, and for this purpose we need some elements of logic. The chief attribute of a mathematical statement is its truth or falsehood. Accordingly, we define the two logical (or boolean) constants TRUE and FALSE, abbreviated T, F or $$1,0$$, respectively. The simplest sentences are the relational expressions, formed by combining pairs of mathematical objects via relational operators. The value of a relational expression is a logical constant. To form more complex sentences we combine relational expressions using logical operators, much like combining numbers using addition and multiplication. The formulation of high-level statements such as (4.1) requires logical functions—called predicates—and quantifiers, which resemble integrals.

These constructs have accompanying symbols, and the symbols of logic are cryptic and alluring. While they help us understand the structure of sentences, they don’t necessarily help us write good sentences. Excessive use of logical symbols clutters the exposition, obscuring meaning. A recurrent theme of this chapter is that, quite often, concepts of great logical depth are better expressed with words.

4.1 Relational Operators

We begin with a simple symbolic sentence:

$$\begin{aligned} 0<1. \end{aligned}$$

(4.2)

This is a relational expression. The relational operator ’$$<$$’ converts two numbers into the logical constant TRUE. Any significant mathematical sentence that evaluates to TRUE is called a theorem, and expression (4.2) is one of the first theorems of analysis; it underpins all inequalities among real numbers.

Relational operators comprise very familiar objects:

$$\begin{aligned} =\qquad \ne \qquad <\qquad \leqslant \qquad >\qquad \geqslant . \end{aligned}$$

(4.3)

These operators are binary, they act on two operands. The first two act on elements of any set; the others act on real numbers (more generally, on elements of an ordered set—see Sect. 4.6).

Interesting things can be said with simple relational expressions:

$$\begin{aligned} 9^3+10^3=1^3+12^3, \quad \qquad \left| \pi -\frac{355}{113}\right| <3\times 10^{-7}. \end{aligned}$$

(4.4)

These expressions also feature prominently in programming languages:

if x $$<$$1 then

x := -x + 1

else

x := x - 1

fi:

Here the logical value of the expression ’x $$<$$ 1’ determines which assignment statement is executed.

At the most basic level we have the relational operators associated with sets, namely the membership and inclusion operators (Sect. 2.1) and their negation:

$$\begin{aligned} \in \qquad \not \in \qquad \subset \qquad \not \subset \qquad \supset \qquad \not \supset . \end{aligned}$$

(4.5)

For example, the relational expression

$$\begin{aligned} \sqrt{2}\not \in \mathbb {Q}\end{aligned}$$

(4.6)

is TRUE, that is, the square root of 2 is not a rational number. We shall prove this fact in Sect. 7.1.

There are countless relational operators in mathematics: the divisibility operator ’$$\vert $$’ and the congruence operator ’$$\equiv $$’ in arithmetic (Sect. 2.1.3), the isomorphism operator ’$$\cong $$’ in algebra, the orthogonality operator ’$$\perp $$’ in geometry, etc. By contrast, the symbol ’$$\approx $$’, used in mathematical physics, does not represent a relational operator because expressions such as $$\pi \approx 3.14$$ cannot be assigned unequivocally a value TRUE or FALSE.