You will probably already seen an introduction to rigorous pure mathematics: some set theory; equivalence relations; and a very brief introduction to some algebraic systems such as the integers, the rationals and the idea of a group. In particular, you will have learnt that every statement, however innocuous it may seem should be proved. There is good reason for this: mathematicians, being human, often make mistakes, and it would be such a shame to base a life-time's work on an erroneous statement made by someone else. The way to weed out any possible mistakes is to write down a watertight proof and check it carefully—and, if possible, have someone else check it for you too.
We are going to look at the mathematical subject
of real analysis, and the properties of
The complexity of the logical arguments does increase when arguments
concerning convergence of sequences are involved. You should be
used to statements of the form for all
.
You now know that such a statement requires a proof (if it is true) or a
counter-example (if false). Similarly, a statement of the form
there exists
requires an example (to show
it is true) or a proper proof (to show it to be false). In analysis, most
statements are more complicated than this, and might take the form
for all
or even there is
, and we need
to understand and work with such statements. This is the point of the
current page.
The phrases for all
and there exists
will be
used so frequently that we adopt special symbols for them: for all
and there exists
.
The variable represents a mathematical object, such as a number,
and the statement that follows (here written as
) is typically
some mathematical property of the object represented by
represents a natural number without us
explicitly saying so, as in
divides
relation only usually makes sense for integers or natural numbers.
Occasionally we will wish to partially specify the domain of the variable
and leave the rest of the specification to be implicit. Thus there will
be many cases where we write for all positive real numbers
.
Example.
The following are all mathematical statements involving quantifiers. You should check that you can read them accurately, including the (implicit or explicit) domain that the variables range over. In some cases, the statements have particularly straightforward informal alternatives in ordinary English; you should check that you see why the formal statement and the informal English one are equivalent.
There are three very important and very basic
things to note about quantifiers and variables before
we can start to use them properly. These are not really theorems, but sort of
rules-of-thumb that you must learn. (In fact, they are closer to being a kind
of mathematical way-of-life that should become completely second nature to you
by the end of this module.) I will introduce them through examples, and then
sum up each one with a moral
afterwards. It is needless to say it, but
not understanding or abiding with these rules will lead to mistakes
of the sort we are trying to avoid.
Example.
For every
This might be the statement
Moral.
A statement with quantifiers may mean something completely different if the domains of the variables are not interpreted correctly. Changing the domain(s) may even turn a true statement into a false statement or vice versa. Forgetting to take note of the domain may lead to an incorrect proof.
Example.
The statements
Moral.
The choice of names of variables in a statement with quantifiers is unimportant. Consistently changing a variable name with another variable name results in the same statement (provided the new name doesn't clash with an existing variable name).
Example.
The statements
Moral.
Changing the order of quantifiers in a statement with more than one quantifier of different types will usually change the meaning of the statement completely.
Any mathematical statement, in particular one involving quantifiers,
will be either true or false; there are no other possibilities.
Much of the work we will do will involve determining which
of these two possibilities is actually the case. A true statement should be proved.
(How to do this is the subject of another web page.) A false
statement has to be proved false. It turns out to be much easier
to turn a false statement
The main advantage of writing our mathematics in formal
mathematical language is that we can then perform operations on the statements we get
(like finding the statement not-
We use the sign not
. This symbol is like a
modified minus sign, and is therefore quite suggestive of not
, so
should be easy to remember.
Proof.
Proof.
These two rules deal conveniently with quantifiers. However a statement
may be built up from more than just quantifiers: it may involve connectives
such as and
(written as or
(written as Implies
(written as and
, or
or Implies
in English
rather than using the symbol for it, that's fine. I will use symbols
and I present further rules to allow you to deal with these when you need to.
If
If
If
The connective Implies
causes many people some trouble as they tend to
think that in a statement
If there are little green men on
Mars then it will rain tomorrow
feels like it should be false, since
the little green men on Mars don't have any control over our weather (we
hope!)
On the other hand, we will see in a moment that the mathematical interpretation
of the statement If there are little green men on
Mars then it will rain tomorrow,
is in fact true
.
Natural language behaves in a very complicated way, and in
one that is very difficult to model mathematically. Much
of the problem is that natural language is not even consistent with itself.
For example, If 1+1=3 then I am a Dutchman
is a correct (i.e.,
true) statement using a popular idiom in English. (But the same idiom does not
work in Dutch, presumably...) So you should forget about all idea of connections
or causality between the two parts of an implication-statement such as
Implies
and just concentrate on whether and
are true or false.
If Implies
only depends on whether and
are true or false it is easy to see that there
are only four possibilities, so we should be able to check each quite
quickly and hence
define what Implies
should mean in each of these four possibilities by
just looking at some examples.
1+1=3 Implies 2+2=6seems to be a true implication, as the second equation is got from the first by multiplying by 2. But both parts are false, so we define
false Implies falseto be true.
1+1=3 Implies 0+0=0seems to be a true implication, as the second equation is got from the first by multiplying by 0. But the second part is now true, so we define
false Implies trueto be true.
1+1=2 Implies 2+2=4seems to be a true implication, as the second equation is got from the first by multiplying by 2. But both parts are true, so we define
true Implies trueto be true.
1+1=2 Implies 2+2=6seems to be a false implication, since we don't expect to be able to deduce any false statement from only true statements. So we define
true Implies falseto be false.
The upshot of this is that Implies
is true if
is false or if is true (or both). Implies
can only be false if is true and is false. So for example
If there are little green men on Mars then it will rain tomorrow
is true
since there are no little green men on Mars and I don't have to be a weather
forecaster to be sure!
If
There is one final point to do with
This means you shouldn't
ever use it at the beginning of a sentence or the beginning of a line.
If you are ever tempted to do this, stop yourself and use the
therefore
symbol
Also, you shouldn't ever use it
in the form
or avoid implies , which in turn implies
.
Note that implies implies
isn't even
grammatically correct English.
All of these rules reduce the problem of negating a mathematical
statement to negating their innermost atomic
parts. We presume you
know how to do that. For example, if
Example.
The statement that a sequence (_{
)
} converges to a limit
So the statement that the sequence (_{ ) } converges to some limit is
And, using the rules in this web page, the statement that the sequence (_{ ) } does not converge to any limit is
The last example is particularly important. Try to follow it in detail, and take plenty of time over this. If you can follow the details you will not have to learn them by rote. (Learning by rote, i.e. without understanding, is to my mind the last resort and is always dangerous because it is easy to make an error. But if you can't follow the arguments you will have to learn the last example and many more like it.)
With all of this background you are in good shape to understand
and work with statements in
analysis such as the sequence (_{
)
} tends to
. The next step is to learn how to prove
(and disprove) such statements. That's the subject of the
next web page.