Ordering Properties - Describing Functions

Mathematical Writing - Vivaldi Franco 2014

Ordering Properties
Describing Functions

Franco Vivaldi1

(1)

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

Franco Vivaldi

Email: f.vivaldi@qmul.ac.uk

We know only a handful of words to describe properties of functions:

injective, surjective, invertible (bijective), constant.

In this chapter we consider attributes of real functions $$f:\mathbb {R}\rightarrow \mathbb {R}$$. We expand our vocabulary substantially, and introduce important terms that are found in more general settings (periodic, bounded, continuous). Then we export this terminology to the real sequences, which are functions defined over $$\mathbb {N}$$.

5.1 Ordering Properties

The real line $$\mathbb {R}$$ is ordered, meaning that for any pair $$(x,y)$$ of real numbers, precisely one of the three relational expressions

$$ x<y \qquad x=y \qquad x>y $$

is true, and the other two are false. We begin with properties that are formulated in terms of ordering.

A simple attribute of a real function is the sign of the values it assumes.

$$\begin{aligned}&\forall x \in \mathbb {R},\,\,f(x)>0 \qquad \,\, {\mathbf {positive}} \end{aligned}$$

(5.1)

$$\begin{aligned}&\forall x \in \mathbb {R},\,\,f(x)<0 \qquad \,\, {\mathbf {negative}} \end{aligned}$$

(5.2)

If (5.1) is formulated with the non-strict inequality $$f(x)\geqslant 0$$, then we say that the function is non-negative. For the inequality $$f(x)\leqslant 0$$, the term ’non-positive’ is uncommon, and one would normally use negative or zero. For example, the exponential function is positive, the absolute value function is non-negative, and the sine function is neither positive nor negative. Because inequalities are reversed under sign change, if a function $$f$$ has any of the stated properties, then $${-}f$$ has the complementary property (e.g., if $$f$$ is positive, then $${-}f$$ is negative).

Note that the terms ’non-negative’ and ’not negative’ have different meaning, the latter being the logical negation of negative. This distinction is very clear in the symbolic definitions.

(The second symbolic expression is obtained by negating (5.2) according to Theorem 4.3.) So a non-negative function is also not negative, but not vice-versa. Two similar expressions with different meanings can easily lead to confusion, and one must remain vigilant.

Next we consider how the action of a function affects the ordering of the real line; the order may be preserved, reversed, or a bit of both. There are two competing terminologies, labelled $$I$$ and $$I\!I$$ in the table below. Each has advantages and disadvantages.

A function that is either increasing or decreasing (strictly or otherwise) is said to be monotonic.

Thus the arc-tangent is increasing in $$I$$ and strictly increasing in $$I\!I$$. In $$I$$ no function can be both increasing and decreasing, and most functions are neither, for instance a constant or the cosine. The constants are the functions that are both non-decreasing and non-increasing.

The disadvantage of $$I$$ is that, as we did above for ’non-negative’, we must differentiate between ’non-increasing’ and the logical negation of increasing ($$\exists x,y\in \mathbb {R},\,\,(x>y) \, \wedge \, (f(x)\leqslant f(y))$$).

The terminology $$I\!I$$ eliminates the annoying distinction between the prefixes ’non-’ and ’not’, but introduces a new problem. Now a constant is both increasing and decreasing, clashing with common usage. (After years without a pay rise, I wouldn’t say: ’my salary is increasing’.)

To be safe, use ’strictly’ to mean strict inequality in any case.