Part 2

New Skills

Formulating Maths
Introduction
Hypotheses &c
Implications
And & or
For all & there exists
Dependence

Proof
Introduction
Counter examples
Constructing proofs
Understanding
Experimentation
Precision
Examples
Assumptions
Contradiction
Induction
The End

 

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Most mathematical statements, in particular most theorems, take the form of making certain assumptions (hypotheses) and drawing certain conclusions (consequences). The hypotheses and/or conclusions may be quite complicated; each may come in several parts. For ease of language, the statement may involve several sentences, and the hypotheses and/or conclusions may themselves be spread over more than one sentence. The statement is true provided that whenever all the hypotheses are satisfied then all the conclusions are satisfied.

For example, consider the following statements:

S1a:

Every positive real number has a unique positive square root.

S1b:

Let x > 0. There exists a unique y > 0 such that y² = x.

Provided that the reader is aware that the context of S1b is real numbers, these statements are synonymous, because, for example, a square root of x is, by definition, a number y such that y² = x, and ``x > 0'' is conventional notation for ``x is a positive real number''. Because S1a and S1b mean the same, it follows (without thinking about the mathematical truth of the statements) that if one of them is a true mathematical statement, then so is the other. Of course, as you well know, both statements are true, but that is a special fact about real numbers. Indeed the following statement, which is very similar to S1a in structure, is false:

S2:

Every positive rational number has a unique positive rational square root.

Thus, the fact that S1a and S1b are true is a fact of mathematics, depending on properties of the real numbers, while the fact that they are synonyms is a fact of language and logic (given the definitions and notation).

Statement S1b is rather more formal than S1a, and the distinction between the hypothesis (x > 0) and conclusion (there exists a unique y > 0 such that y² = x) is clearer in S1b; nevertheless, S1a has the same hypothesis as S1b, and the same conclusion as S1b. Moreover, S1a has the merit of mentioning explicitly the context of real numbers. Which you choose to write is mainly a matter of taste and context.

You should be aware that the conclusion of statement S1b (or S1a) actually says two things:

(i)

there exists at least one y > 0 such that y² = x (existence),

(ii)

there exists at most one y > 0 such that y² = x (uniqueness).

Statements S1a and S1b are not only true statements, they are theorems, or more precisely two equivalent statements of the same theorem. You may be surprised about this: the existence of square roots of positive real numbers is not a law of nature, but is something which can be deduced from a few axioms about the system of real numbers. (Most of the axioms are simple laws of arithmetic, e.g. a(b+c) = ab+ac, or laws concerning the ordering, e.g. if a £ b and 0 £ c, then ac £ bc; however, there is one more complicated axiom which distinguishes the real numbers from the rational numbers, say-see Alice in Numberland, Chapter 8, for example.)

To show that S1a and S1b are true statements, it is necessary to prove the theorem. Since the statements have the same meaning, it is sufficient to prove one of them. Since the conclusion has two parts, (i) and (ii) above, the proof will also have two parts. The proof of (i) depends on the complicated axiom of the real numbers, and we will not give it here. However, the proof of (ii) is quite short, and uses only the simple axioms, or familiar properties of real numbers easily derived from the axioms, so we will give the proof here, for the interest of readers who may be becoming bored by the absence of mathematics.

Suppose that y₁ > 0, y₁² = x, y₂ > 0, and y₂² = x. [Our objective is to show that y₁ = y₂, which will complete the proof of (ii).] Then

0 = x-x = y12 - y22 = (y1-y2)(y1+y 2).

But y₁+y₂ > 0 (since y₁ > 0 and y₂ > 0). Since y₁+y₂ ¹ 0, we may divide by y₁+y₂, and deduce that 0 = y₁-y₂, so y₁ = y₂, as required.

To show that S2 is false, it is sufficient to give one example which satisfies the hypothesis of S2, but not the conclusion. In other words, it is sufficient to exhibit one positive rational number x which does not have a (unique, positive) rational square root. This can be achieved by taking x = 2. For completeness, we should prove that 2 does not have a rational square root (in common parlance, that Ö2 is irrational), but the reader probably knows how to do this, and the argument was given in Section 2.1.

Another aspect of S1b which merits comment is the use of the word ``Let'' in the phrase ``Let x > 0''. This is shorthand notation for ``Let x be a positive real number'', but what does this mean? It tells the reader that we are going to consider one positive real number, which we will call x, but we do not know which positive real number number it is. In other words, x is an arbitrary positive real number x. Similarly, the statement

S3:

Let x be a real number such that 0 < x < 1

tells the reader that we are going to consider one real number which we will call x, that x lies (strictly) between 0 and 1, but that x is otherwise arbitrary.

Consider the statements:

S4a:

Let R be the set of real numbers;

S5a:

Let A be a set of real numbers.

These statements are almost identical, but their meanings are different; S4a means that R is the set of all real numbers, while S5a means that A is a set of some real numbers. The clue here is that the use of ``the'' in S4a, indicating that we are discussing a unique set, while ``a'' in S5a indicates that A is one of many possible sets. Actually, instead of S4a and S5a, it would be better to write:

S4b:

Let R be the set of all real numbers;

S5b:

Let A be a subset of R.

[The notation R is standard for the set of all real numbers.]

We have already pointed out that the hypotheses of a statement may be spread over several sentences. One particular trap that may catch you out is that sometimes a standing hypothesis is made, which is not repeated in every theorem. For example, consider the following statement:

S6:

Suppose that f(0) < 0 and f(1) > 0. Then there exists a such that 0 < a < 1 and f(a) = 0.

When you read S6, it is natural to assume that f is a real-valued function of a real variable between 0 and 1, i.e. that f specifies a real number f(x) for each value of x with 0 £ x £ 1. However, if that is all that you assume, apart from the hypotheses f(0) < 0 and f(1) > 0 which are given explicitly, then S6 is false. Indeed, one may take

f(x) = -1    if (0 £ x £ 1 2 )    or 1 if ( 1 2 < x £ 1).

Clearly, f satisfies the hypotheses of S6, but not its conclusion. Thus, S6 is false.

However, suppose that the writer of S6 had at some previous stage, maybe many pages earlier, written: From now on, f will be a continuous function. Then S6 would be a true statement. This is a theorem which is sufficiently deep to have a name (the Intermediate Value Theorem), and the proof is beyond us here. It is one of those theorems which appears obvious (if you have to get from below the x-axis to above it with a continuous curve, then you have to cross the x-axis somewhere-see the diagram on the right), but one cannot even begin to prove it until one has made a precise definition of continuous.

This shows that it is very important to identify all the hypotheses in a statement. You will also need to distinguish correctly between the hypotheses and the conclusions, both when reading mathematics, and when writing. This may sound like a simple task, but there are many pitfalls surrounding the word ``if'', which we shall discuss in thenext section.