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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Two other phrases which occur frequently in mathematics are ``for all'' (and its synonyms) and ``there exists'' (and its synonyms). The typical structure of a statement containing them is:

S31a:

For all x Î S, P(x) holds.

S32:

There exists x Î S such that P(x) holds.

Here, S is some set, Î is the standard set-theoretic symbol denoting either ``belonging to'' or ``belongs to'', and P(x) is a statement which depends on some parameter x. For example, if S is the set of all real numbers, and P(x) is the statement x² = 2, then S31a becomes the false statement:

S33a:

For all real numbers x, x² = 2,

and S32a becomes the true statement:

S34a:

There exists a real number x such that x² = 2.

Often, the distinction between ``for all'' and ``there exists'' is very clear, for example in S33a and S34a, and in the following:

S35:

All undergraduates are female;

S36:

There exist undergraduates who are female.

However, with complicated mathematical statements, mistakes are sometimes made, so you need to be careful when reading. Moreover, you cannot expect the reader to guess which quantifier you mean, so you should always include them (or some equivalent wording) in your written work.

The phrases ``for all'' and ``there exists'' (and their synonyms) are known as quantifiers; they are often denoted by the symbols " and $, respectively. In S31a and S32a, the variable x is said to be subordinate to the quantifier; the set S is known as the range of the quantifier. Any statement containing the quantifier " or $ should include a subordinate variable and a range. You are recommended not to use these symbols as general abbreviations for ``all'' and ``exists''.

In both S33a and S34a, the variable is x and the range is the set R of all real numbers. Thus, S33a and S34a may be rewritten as:

S33b:

["x Î R] x² = 2,

S34b:

$x Î R such that x² = 2.

Phrases which may be synonymous with ``for all'' include

Given,     whenever,     for any,     every,     foreach, etc.

Phrases which may be synonymous with ``there exists'' include

for some,     for at least one,     there is,     has, etc.

It is also possible to rewrite S31a in an ``If ..., then ...'' format (see Implications for further reformulations):

S31b:

If x Î S, then P(x) holds.

Note that in statements S31a and S32, x is merely a dummy variable, and can be replaced by any other symbol representing a variable, provided that this is done consistently and provided that the new symbol is not used for any other purpose. Thus, S31a is synonymous with:

S31c:

For all y Î S, P(y) holds.

When writing mathematics, it is important to observe the following general principle:

The name (i.e. symbol) given to a variable subordinate to a quantifier may be chosen arbitrarily, provided that the name chosen is not already in use in the argument.

It is particularly important to observe this principle in the case of the quantifier $, when all sorts of logical chaos is likely to break out if the principle is not observed. For example, the following is a true statement:

S37:

For any integer n ³ 2, there is a prime p which divides n.

However, the following argument becomes gobbledygook:

S38:

Let n be any integer with n ³ 2, and let p be any prime. By S37, there is a prime p which divides n.

At this point, there is a danger that you will deduce that the arbitrary prime p divides n, i.e. that n is divisible by all primes! Instead of S38, one should write:

S39:

Let n be any integer with n ³ 2, and let p be any prime. By S37, there is a prime q which divides n.

S39 is true, and clear, even if it is not very interesting.

In advanced mathematics, one often comes across statements with several quantifiers embedded inside each other. For example, the following is a typical mathematical statement:

S40a:

Given e > 0, there exists d > 0 such that |f(y)-f(x)| < e whenever |y-x| < d.

S40a includes one explicit ``there exists'' and two implicit ``for all''s, signalled by ``given'' and ``whenever''. We can write S40a in the following formal fashion:

S40b:

"e > 0 $d > 0 such that " ( (x,y) such that |y-x| < d)     |f(y)-f(x)| < e.

In S40b, it is presumed that f is a function (of one variable) which has already been introduced; e and d, and the pair x,y are variables which are subordinate to quantifiers, so they can be replaced by other symbols if you wish, provided this is done consistently. The range of e, and the range of d, are both the set of all positive real numbers; the range of the pair x,y is the set of all pairs of real numbers satisfying |y-x| < d.

In fact, in saying that S40a and S40b are synonymous, we have cheated slightly. This is correct, provided that neither x nor y has already been introduced. In that case, they are both subordinate to the quantifier ``whenever'', and rewriting S40a as S40b is correct. However, if x had been introduced at some stage before S40a is stated (but y had not been), then x cannot be subordinate to the quantifier. In that case, S40a should be rewritten as:

S41:

"e > 0 $d > 0 such that " (y such that |y-x| < d)     |f(y)-f(x)| < e.

This apparently small change affects the meaning of the statement significantly. We shall return to this subtle, but important, point in the next section.