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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Consider the following statement:

S42:

"x > 0   $y > 0 such that y² = x.

Is it synonymous with the following statement?

S43:

$y > 0 such that "x > 0     y² = x.

Of course not! When you convert the symbols into English, it should be clear that S42 is the true statement that every positive number has a positive square root. However, S43 is the nonsensical statement that there is one positive number whose square equals all positive real numbers simultaneously.

This example shows that in statements involving quantifiers, the order in which the variables are introduced matters crucially. This is not really a point of mathematics at all, merely one of languauge, but it is of great significance for mathematicians, and it often causes difficulty for undergraduates.

In S42, the clause $y > 0 appears after the introduction of the variable x. It therefore follows from the structure of the sentence (``For all x, there exists y such that ...'') that the variable may depend on x. Similarly, in S40a, S40b, and S41, d may depend on e. On the other hand in S43 (``There exists y such that for all x ...''), y cannot possibly depend on x, because x has not even been introduced to the reader's mind when y is supposed to be determined. We can summarise this principle:

A variable subordinate to a quantifier $ (``there exists'', ``for some'', etc.) may depend on variables which have been introduced earlier.

For example, consider the statement:

S44a:

Every undergraduate in my college owns a bicycle.

This can be formalised as:

S44b:

"x Î {undergraduates in my college}   $y Î {bicycles} such that x owns y.

Of course, the bicycle y depends on the undergraduate x. Since the undergraduate is introduced in the statement before the bicycle, this is consistent with the principle. On the other hand, in the statement:

S45:

Every college has a library which is available for all students in the college,

the library depends on the college (introduced earlier) but not on the student (introduced later in the statement).

This principle applies even when the other variables have been introduced in earlier sentences, including those which appeared considerably earlier. For example, consider the following statement (compare S40a and S41):

S46:

Let x Î R. A function f: R ® R is said to be continuous at x if, for all e > 0, there exists d > 0 such that |f(y)-f(x)| < e whenever |y-x| < d.

Here, d may depend on x, f and e (all previously introduced), but not on y (introduced later). On the other hand, in the statement:

S47:

Let f: R ® R be a function. Then f is said to be uniformly continuous if, for all e > 0, there exists d > 0 such that |f(y)-f(x)| < e whenever |y-x| < d,

d should be independent of both x and y (not previously introduced), but d may depend on f and e.

To avoid ambiguity, and to stress what depends on what, you may find that authors write a statement such as S40a in the form:

S40c:

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

This is a notational device which can be helpful in reminding both the writer and the reader that d may depend on e. Sometimes, the author might mention explicitly that, in S47, d is independent of x and y. These are good practices to adopt yourself in your notes and written solutions, at least until you are confident that you will not get confused.

Incidentally, understanding why S46 is a sensible definition of continuity is outside the theme of these notes, but you might like to think about it (see Alice in Numberland, Chapter 11, for example).