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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Suppose that we wish to formulate a statement in which we make some hypotheses and draw certain conclusions from them. Let P stand for all the hypotheses and Q for all the conclusions. For example, in S1b, P stands for: x > 0 and Q for: there exists a unique y > 0 such that y² = x. We can then write our desired statement as:

S7a:

If P, then Q.

Thus we can rewrite S1a and S1b in the format of S7a:

S1c:

If x > 0, then there exists a unique y > 0 such that y² = x.

In principle, any statement with a set of hypotheses and a set of conclusions can be written in the form S7a, but in practice it may be clumsy to do this, especially if P and/or Q is complicated. In this section, we are going to discuss logical aspects of statements containing the word ``if''.

As we mentioned in Hypotheses and conclusions, the statement S7a is true provided that whenever the hypotheses P are satisfied, then the conclusions Q are satisfied. In mathematics, to avoid ambiguity we have to stick rigidly to this principle. This differs from everyday conversation, where people often do not distinguish carefully between S7a and its converse:

S8a:

If Q, then P.

A synonymous way of expressing the converse is:

S8b:

If P does not hold, then Q does not hold.

(The statement P does not hold or P is false is known as the negation of P; we shall have more to say about negations in Proofs by contradiction.)

For example, a statement such as:

S9:

If you get three As in A Level, I'll buy you a computer

is often taken to include the converse (if you don't get three As, I won't buy you a computer). In fact, strictly speaking (and mathematicians often have to speak strictly), there is no reason to infer the converse from S9, so if someone did make the statement S9 to you and you did not get three As, you are quite entitled to ask them whether they are going to buy you a computer. On the other hand, if someone made this statement and you did get three As, then you are entitled to demand your computer.

Since mathematics is a very precise subject, we do have to be very careful what is meant when reading, and more particularly when writing, mathematics. So, we can write:

S10:

If x > 1, then x² > 1

(a true statement), but we should not get confused and write the converse:

S11:

If x² > 1, then x > 1

(a false statement). Equally, we should not confuse the following two statements, which are converses of each other:

S12:

If x > 0, then there exists y > 0 such that y² = x.

S13:

If there exists y > 0 such that y² = x, then x > 0.

Both S12 and S13 are true, but they have different meanings.

There are many statements which have the same meaning as S7a. For example, the following are all synonyms:

S7a:

If P, then Q.

S7b:

P implies Q.

S7c:

Let P hold. Then Q holds.

S7d:

P only if Q.

S7e:

P is a sufficient condition for Q (or: In order that Q holds, it is sufficient that P holds).

S7f:

Q is a necessary condition for P (or: In order that P holds, it is necessary that Q holds).

S7g:

If Q does not hold, then P does not hold.

When making a mathematical statement, the choice of format for the statement is up to the writer, but one should try to choose a style which is readable, clear, and unambiguous. Depending on the context, some of the formats listed above are likely to be unsuitable, and there may be convenient formats which are not listed.

To revert to a particular example, the following are all synonymous with S1a:

S1a:

Every positive real number has a unique positive square root.

S1c:

If x > 0, then there exists a unique y > 0 such that y² = x.

S1b:

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

S1d:

x > 0 only if there exists a unique y > 0 such that y² = x.

S1e:

x > 0 \implies there exists a unique y > 0 such that y² = x.

S1f:

In order that there exists a unique y > 0 such that y² = x, it is sufficient that x > 0.

S1g:

A necessary condition that x > 0 is that there exists a unique y > 0 such that y² = x.

S1h:

If there does not exist a unique y > 0 such that y² = x, then x £ 0.

Take your pick! Both S1d and S1g are very peculiar ways to make this statement, and you are unlikely to choose them.

We have mentioned above that S7a is synonymous with S7d. Interchanging the roles of P and Q, we see that

S8c:

Q only if P

is synonymous with S8a. Thus S8c is the converse of S7a. In short, ``only if'' is the converse of ``if''. In practice, the phrase ``only if'' is rarely used on its own, but it is often used in the phrase ``if and only if''.

If one wishes to make a statement ``If P, then Q'' and its converse ``If Q, then P'' simultaneously, one may write

S14:

P if and only if Q.

(occasionally, you may see this abbreviated to ``P iff Q''). Thus S14 is really two statements. So, to prove S14, you will usually have to give two separate proofs, one of ``If P, then Q'' and one of ``If Q, then P''; only rarely is it possible to prove both parts simultaneously (see also Proof by contradiction). Note that S14 is synonymous with ``Q if and only if P''.

For example, we could combine the statements S12 and S13 into a single statement:

S15:

There exists y > 0 such that y² = x if and only if x > 0.

However, S15 could be ambiguous; it might mean (as it is supposed to):

S16a:

[There exists y > 0 such that y² = x]     if and only if     [x > 0],

or it might mean:

S17:

There exists y > 0 such that     [y² = x if and only if x > 0].

The reader could probably work out that we do not mean S17, either from the context, or by observing that S17 is false (S17 implies that there is some positive number whose square equals all positive numbers, which is clearly nonsense). Nevertheless, we should try to avoid ambiguity if it is possible to do so without becoming unreadable, so instead of S15, it would be better to write S16a or one of the following:

S16b:

x > 0 if and only if there exists y > 0 such that y² = x,

S16c:

A necessary and sufficient condition that there exists y > 0 such that y² = x is that x > 0,

S16d:

The following are equivalent:

     (i) x > 0,

    (ii) there exists y > 0 such that y² = x.

The format of S16d is most appropriate for complicated statements, or if there are more than two equivalent statements ((i),(ii),(iii),...).

In Writing mathematics, we discussed the need to choose language and punctuation in such a way as to remove ambiguity. One useful tip is that any statement starting with ``If'' should have a matching ``then''. This makes it clear when the hypotheses are completed and the conclusions start. For example, what does the following mean?

S18:

If n is prime, n ¹ 2, n is odd.

It could mean:

S19:

If n is prime, then n ¹ 2 and n is odd,

or

S20:

If n is prime and n ¹ 2, then n is odd.

Of course, S19 is false and S20 is true, so you (an educated mathematician) know which interpretation to take. However, there is no point in only making statements whose truth is obvious to the reader (it frequently happens that mathematicians find statements which they wrote themselves are far from obvious after an interval in which they have forgotten the context). An uneducated person might misinterpret S18, by taking it to mean S19 or even

S21:

If n is prime or n ¹ 2, then n is odd,

or

S22:

If n is prime, then n ¹ 2 or n is odd.

So the moral is that you should include all those little words like ``then'', ``and'', ``or'', etc.