- Mathematical statements have very precise
- Typically, a mathematical statement has certain
hypotheses and certain conclusions. The statement
is true if, whenever the hypotheses are satisfied,
then the conclusions are satisfied.
- A hypothesis such as ``Let x be a whatsit''
means that x is one of many whatsits, but we do not
specify which whatsit.
- The hypotheses of a statement may be spread
over several sentences, and may include standing
hypotheses made a long time earlier.
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
For example, consider the following statements:
- Every positive real number has a unique
positive square root.
- 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:
- 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
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
You should be aware that the conclusion of statement
S1b (or S1a) actually says two things:
- there exists at least one y > 0 such
that y2 = x (existence),
- 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
, 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
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.
|0 = x-x =
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
that 2 does not have a rational
square root (in common parlance, that Ö
2 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
- Let x be a real number such that 0 < x <
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:
- Let R be the set of real
- Let A be a set of real numbers.
These statements are almost identical, but their
meanings are different; S4a means that R
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:
- Let R be the set of all real
- Let A be a subset of R.
[The notation R
is standard for the set of all
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
- Suppose that f(0) < 0 and f(1) > 0. Then
there exists a such that 0 < a < 1 and f(a) =
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
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 £
or 1 if (
|< 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.