Pure Mathematics, Lecture Notes - Mathematics 1, Study notes of Mathematics

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INTRODUCTION TO PURE MATHEMATICS

R. W. Knight

Michaelmas 2010

1. Beginning mathematics

1.1. Introduction

There are various ways to define mathematics. One way (which, like the others, has its strengths and weaknesses) is that we can define mathematics by its distinctive epistemology: that is, the methods used to determine the statements that we will regard as true. The epistemology of mathematics is uniquely powerful. We are used to hearing of the received wisdom in physics, history etc. being overturned by new evidence. This is not our experience in mathematics. Learning mathematics involves—indeed, to a certain extent learning mathematics is— learning how mathematicians establish mathematical truth: in learning how to do proofs. A statement which has a proof is called a theorem.

So mathematics is built out of theorems which have proofs. It is impossible to under- stand mathematics without understanding how proofs work.

An example:

Theorem 1.1.1. There are infinitely many prime numbers.

Proof: Suppose that there are only finitely many prime numbers. List them as p 1 , p 2 ,... , pn.

Let P = p 1 p 2... pn + 1. Then P is a positive integer greater than 1, so it has a prime factor; call this p. Now p, being a prime number, must be one of p 1 , p 2 ,... , pn. But then, dividing P by p gives a remainder of 1, so p is not a factor of P after all.

This is a contradiction. So our original assumption, that there are only finitely many prime numbers, is impossible; and the result follows. 

The proof establishes the theorem beyond doubt; it is impossible for a rational person (at any rate, one who knows a minimum amount about prime numbers) to read the proof and not believe the theorem. This is different from our experience in other subjects. Rational people can disagree about God, what finished off the trilobites, or the Trojan War. They do not disagree about how many prime numbers there are.

While our detailed standards of what counts as a proof have changed somewhat, our ideal has not changed in over two thousand years: in principle, a proof establishes the truth of a theorem beyond dispute, for ever.

This concept of a proof is largely responsible for the power and success of mathematics. In the later part of this course, we examine, briefly, how proofs work, and how to construct them. But the proof just given also shows us that mathematics is a language. We talked about “prime numbers”. We used notation, such as +, = and pi. Proofs must be expressed in an appropriate language; and we will be looking at this in the first part of the course. Indeed, pursuing the analogy of language, you could say that we will begin by looking at some basic words and grammar; and move on, next week, to examine discourse structure: the legitimate ways of structuring what we say in our language.

One particularly important set: Definition 2.1.2. The empty set, written ∅, is the set with no elements. Make sure your empty sets ∅ look different from the Greek letter phi φ. Some more useful notation: Notation 2.1.3. The set whose elements are a 1 , a 2 , a 3 ,... , an is written {a 1 , a 2 ,... , an}, with curly brackets. If a 1 , a 2 , a 3 ,... are an infinite sequence of objects, we can write the set containing them as {a 1 , a 2 , a 3 ,.. .}. An error to avoid:

 Do not confuse a and {a}, the set whose only element is a. For example, if a = ∅, then a

has no elements, but {a} has one element (namely a), so they can’t be equal. Or, if a is the set of all real numbers, then a is infinite, but {a} isn’t. And so on. Examples 2.1.4. { 0 , 1 } and { 1 , 0 } have the same elements, so they are equal. Similarly, { 2 , 2 } and { 2 } have the same elements, and so are equal. We also have notation for when the members of a set are defined by a property. Notation 2.1.5. Suppose P (x) is a property. Then {x : P (x)}, also written {x | P (x)}, is the set of all objects x having the property P (x). If A is a set, and P (x) is a property than an element x of A might possess, then we write {x ∈ A : P (x)} or {x ∈ A | P (x)} for the set of elements x of A having the property P.

Examples 2.1.6. The set of even numbers is {n ∈ Z : n is even}. We could write the set of primes as

{n | n is a prime number},

or as {n ∈ N | n is prime}.

The set { 1 , 2 , 3 , 4 , 6 , 12 } is equal to {n ∈ N : n is a factor of 12 }. Some other important sets: Notation 2.1.7. N is the set { 0 , 1 , 2 , 3 ,.. .} of all natural numbers. Z is the set of all integers (both positive and negative, and zero). (Z is the first letter of the German word Zahlen “numbers”.) Q is the set of all rational numbers. (Q for quotient.) R is the set of all real numbers. R+^ is the set of all positive real numbers, and R−^ is the set of all negative real numbers. C is the set of all complex numbers. All the above are written in the blackboard bold font, which was originally a way of writing bold-face letters on a blackboard. It has since taken on an independent life. One gremlin in the above list: some books and authors include 0 in the set N, and some don’t. There is, unfortunately, no standard convention. In this course, 0 will belong to N; in others it may not do.

Definition 2.1.8. A subset I of R is an interval if and only if, whenever x, y ∈ I, and x < z < y, then z ∈ I. The various types of intervals are written as follows. Suppose that a and b are real numbers, and a ≤ b.

1. (a, b) is the set of all real numbers c such that a < c < b; more compactly, (a, b) = {c ∈ R : a < c < b}.
2. [a, b] = {c ∈ R : a ≤ c ≤ b}.
3. (a, b] = {c ∈ R : a < c ≤ b}.
4. [a, b] = {c ∈ R : a ≤ c < b}.
5. (a, ∞) = {c ∈ R : a < c}.
6. [a, ∞) = {c ∈ R : a ≤ c}.
7. (−∞, b) = {c ∈ R : c ≤ b}.
8. (−∞, b] = {c ∈ R : c < b}.
9. (−∞, ∞) = R. We have notation for measuring the sizes of sets.

Definition 2.1.9. Suppose A is a set with n elements (where n is a natural number). Then we write |A| = n, and say that n is the cardinality or size of A.

Measuring the sizes of infinite sets will be touched on the Analysis course.

2.2. Set algebra

Definition 2.2.1. If A and B are sets, then A is a subset of B, which we write A ⊆ B, if and only if every element of A is an element of B. If in addition A 6 = B, then A is a proper subset of B, and we write A ⊂ 6 =

B.

We can also write A ⊆ B as B⊇A. We often reverse the sense of a relation by writing it backwards. Thus a < b means the same as b > a, a ∈ A means the same as A ∋ a, etc.

Examples 2.2.2. N ⊆ Z ⊆ Q ⊆ R ⊆ C. [− 1 , 1] 6 ⊆[0, 2].

We often negate a relation by drawing a line through it. Thus we write a /∈ A to mean “it is not true that a ∈ A”, a 6 ≤ b to mean “it is not true that a ≤ b”, and so on.

Theorem 2.2.3. Suppose A and B are sets. Then A = B if and only if A ⊆ B and B ⊆ A.

Proof: If A = B, then A and B have the same elements, so every element of A is an element of B and every element of B is an element of A. If A ⊆ B and B ⊆ A, then every element of A is an element of B and every element of B is an element of A, so A = B. 

Some familiar set operations:

Definition 2.2.4. Suppose A and B are sets. The intersection of A and B is

A ∩ B = {x : x ∈ A and x ∈ B}.

The union of A and B is

A ∪ B = {x : x ∈ A or x ∈ B}.

2.3. Ordered pairs

Definition 2.3.1. The ordered pair whose first element is a and whose second element is b, is written (a, b). (a, b) = (c, d) if and only if a = c and b = d. In other words, given an ordered pair (a, b), one of a and b is the first, and the other is the second. This is in contrast to the unordered pair {a, b}, where you cannot identify a first and second element; {a, b} and {b, a} have the same elements, and so they are equal.

 There is a problem with notation here. If a and b are real numbers and a < b, then the

ordered pair whose first element is a and whose second element is b, and the open interval between a and b, are both written (a, b). Don’t get confused! If, in your work, there is the possibility of confusion, then remove the ambiguity by writing things like “the open interval (a, b)”, or “the ordered pair (a, b)”. We could of course change our notation: for example, we could write the ordered pair as 〈a, b〉 or the open interval as ]a, b[. But no such scheme is accepted by everybody. One reason for this is that there are more mathematical concepts than notations to represent them; so ambiguities like this, unfortunately, are frequent. We can also define ordered triples (a, b, c), ordered quadruples (a, b, c, d), etc. in the same manner. A sequence n long is called an n-tuple. Definition 2.3.2. The Cartesian product A × B of two sets A and B is the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B. If A = B, we also write A × A as A^2. More generally, we define A 1 × A 2 × · · · × An to be the set of all ordered n-tuples (a 1 , a 2 ,... , an) such that for all i, ai ∈ Ai. The product of n copies of A, we write as An. (A^1 is A; and A^0 is {∅}.) Examples 2.3.3. The most familiar example of a Cartesian product is the Euclidean plane R^2 = R×R, which we regard as being equal to the set of ordered pairs of real numbers. If A = { 1 , 2 } and B = { 3 , 4 , 5 }, then A × B =

while B × A =

 Notational problem: the elements of the Cartesian product A × B are ordered pairs (a, b).

They are not written a × b. Theorem 2.3.4. Let A, B, C and D be any sets. Then

(A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).

Proof: ⊆: Suppose (x, y) ∈ (A × B) ∩ (C × D). Then (x, y) ∈ A × B and (x, y) ∈ C × D. Since (x, y) ∈ A × B, x ∈ A and y ∈ B. Similarly x ∈ C and y ∈ D. So x ∈ A ∩ C and y ∈ B ∩ D. So (x, y) ∈ (A × B) ∩ (C × D). ⊇: Suppose (x, y) ∈ (A ∩ C) × (B ∩ D). Then x ∈ A ∩ C and y ∈ B ∩ D. So x ∈ A and y ∈ B, so (x, y) ∈ A × B.

Similarly (x, y) ∈ C × D. So (x, y) ∈ (A × B) ∩ (C × D). 

Notation 2.3.5. Suppose that to each element i of some indexing set I some object ai is assigned. Then the set of all the ai, for i an element of I, is written

{ai : i ∈ I},

and is an indexed family.

Example 2.3.6. Suppose that, for each i ∈ N \ { 0 }, pi is the ith prime number. Then

{pi : i ∈ N \ { 0 }}

is the set of all prime numbers.

Definition 2.3.7. Suppose that I is an indexing set, and {Ai : i ∈ I} is an indexed family consisting of sets. Then the union of all the Ai, for i ∈ I, is the set ⋃

i∈I

Ai = {x : ∃i ∈ I x ∈ Ai}.

If I is non-empty, then the intersection of all the Ai, for i ∈ I is the set ⋂

i∈I

Ai = {x : ∀i ∈ I x ∈ Ai}.

∀ is read as “for all”, and ∃ is read as “there exists”.

3. Relations and Functions

3.1. Relations

Intuitively, a binary relation is something like =, ≤ or ⊆ which asserts some relation- ship between two objects. We formalise this idea by indentifying a relationship a R b with the set of ordered pairs (a, b) which are connected by the relation.

Definition 3.1.1. A binary relation between two sets A and B is a subset of A × B. If A is a set, a binary relation on A is a subset of A × A. If R is a relation, we write (a, b) ∈ R and a R b interchangeably.

Examples 3.1.2. The order relation on the set R of real numbers is the set {(a, b) ∈ R × R : a ≤ b}.

We will be meeting many kinds of relations. But one of the most important kinds of relation is the equivalence relation, which asserts that two objects are, in some sense, the same.

Definition 3.1.3. Let R be a relation on a set A. R is reflexive if and only if, for all a ∈ A, a R a.

We prove that PR is a partition. If P ∈ PR, then for some a ∈ A, P = [a]. By reflexivity of R, a R a. Hence a ∈ [a]; so a ∈ P , so P 6 = ∅. So ∅ ∈/ PR. Suppose that a ∈ A. Then by reflexivity of R, a R a. So a ∈ [a]. Hence a ∈

b∈A[b]; that is, a ∈

P ∈PR P^. Hence^ A^ ⊆^

P ∈PR P^. On the other hand, if^ P^ ∈^ P, then for some a ∈ A, P = [a] = {b ∈ A : a R b}; so P ⊆ A. Hence

P ∈PR P^ ⊆^ A. Hence^

P ∈PR P^ =^ A.

Now suppose that P, Q ∈ PR, and P ∩ Q 6 = ∅. Suppose that a ∈ P ∩ Q. Since P, Q ∈ PR, there exist b, c ∈ A such that P = [b] and Q = [c]. Suppose d ∈ Q. Then d ∈ [c], so b R c. Also a ∈ Q, so a ∈ [c]. So c R a. By symmetry of R, a R c. By transitivity of R, a R d. Now a ∈ P , so a ∈ [b]. So b R a. By transitivity of R, b R d. So d ∈ [b], that is, d ∈ P. So Q ⊆ P. Repeating the above argument but interchanging the roles of P and Q, we prove that P ⊆ Q. Hence P = Q, and PR is a partition. Now let P be any partition. Define a relation RP so that a RP b if and only if there exists an element P of P such that a, b ∈ P. Let a be any element of A. Now A =

P ∈P P^ , so there exists an element^ P^ of^ P such that a ∈ P. Then a, a ∈ P. Hence a RP a. So RP is reflexive. Suppose that a RP b. Then there exists P ∈ P such that a, b ∈ P. Then b, a ∈ P. So b RP a. So RP is symmetric. Suppose that a RP b and b RP c. Then there exist P and Q in P such that a, b ∈ P and b, c ∈ Q. Then b ∈ P, Q, so b ∈ P ∩ Q, so P ∩ Q 6 = ∅. So P = Q, and now a, c ∈ P. So a RP c. So RP is transitive. 

Exercise 3.1.9. Show that RPR = R and PRP = P.

3.2. Functions

The concept of a function from one set A to B is very familiar: it is a rule assigning exactly one element of B to each element of A. We formalise the concept as follows:

Definition 3.2.1. A function from a set A to a set B is a set f of ordered pairs from A × B such that for all a ∈ A, there exactly one b ∈ B such that (a, b) ∈ f. We write f (a) = b. If f is a function from A to B, we refer to A as its domain and B as its codomain. The set of values {f (a) : a ∈ A} is the range of f.

Two functions f and g from A to B are equal if and only if for all a ∈ A, f (a) = g(a). Mild health warning: mathematicians use the terminology of ranges and codomains somewhat inconsistently.

Notation 3.2.2. If f is a function from A to B, we write f : A → B.

There is a similar notation for defining functions. As an example, if f is the function from R to R taking each real number to its square, we write the definition of f thus:

f : x 7 → x^2.

It may not be clear that a recipe for defining a function makes sense. For example, if we are defining a function from R to R, then the recipe

f : x 7 →

x

fails if x = 0; while the recipe

f : x 7 → y where y^2 = x

does not return a unique value: is f (4) equal to 2 or −2?

In either case, where either f (x) cannot always be defined, or where f (x) appears to take more than one value, there is something wrong with the definition: we say that f is ill-defined. If f is not ill-defined, then it is well-defined.

Some special kinds of functions:

Definition 3.2.3. A function f from A to B is:

1. one-to-one, or injective, or an injection, if for all a and a′^ in A, f (a) = f (a′) only when a = a′.
2. onto, or surjective, or a surjection, if for all b ∈ B, there exists a ∈ A such that f (a) = b.
3. one-to-one and onto, or bijective, or a bijection, if and only if it is both injective and surjective.

Examples 3.2.4. Our examples will be functions from R to R.

1. The function f : x → x^2 is not injective, because f (1) = f (−1) while 1 6 = − 1. It is not surjective either, because − 1 is not equal to f (x) for any x.
2. The function g : x → ex^ is one-to-one. However it is not surjective, because again − 1 is not in the range of g.
3. The function h : x → ex^ sin x is onto. However it is not one-to-one, because, for instance, h(0) = h(2π), while 0 6 = 2π.
4. The function k : x → x^3 is both one-to-one and onto, so it is a bijection. One particularly important function:

Definition 3.2.5. Let A be any set. The identity function on A is the function idA from A to itself defined so that idA(a) = a for all a ∈ A.

We define the image f (X) of X under f to be the set {f (x) : x ∈ X}. We define the preimage f −^1 (Y ) of Y under f to be the set {x : f (x) ∈ Y }.

 There is a serious possibility of notational confusion here. If X ⊆ A, and x ∈ A, then

f (X) and f (x) look very similar, even though they’re different kinds of object: the first is a set, the second a single value. It’s even worse with f −^1 (Y ); for this can sensibly defined even if f is not one-to-one, and f −^1 doesn’t exist as a function. Some people get around this by writing images and preimages with square brackets: f [X] and f −^1 [Y ]. Definition 3.3.7. Suppose f : A → B, and X ⊆ A. Then the restriction f ↾X of f to X is the function from X to B defined so that for all x ∈ X,

f ↾X (x) = f (x).

So the only difference between f and f ↾X is that f ↾X is sometimes undefined. Example 3.3.8. Suppose f : R → R is defined so that f (x) = x^2 for all x ∈ R. Then f ↾R+ is the function from R+^ to R so that for all x ∈ R+, f ↾R+ (x) = x^2. Thus f ↾R+ (2) = f (2) = 4; f (−2) is also 4 , but f ↾R+ (−2) doesn’t exist. Observation 3.3.9. If f : A → B and X ⊆ A, then the image f (X) of X under f is equal to the range of f ↾X. Restrictions are written in various ways. You will see f ↾X written as f ↾X, f |X , f |X etc.

4. Mathematical Induction

4.1. The Principle of Induction

We now revisit the method of Mathematical Induction: Theorem 4.1.1. (Principle of Mathematical Induction) Suppose P is a property of natural numbers, that P (0) is true, and that for all natural numbers n, if P (n) is true, then P (n + 1) is true also. Then P (n) is true for all natural numbers n. Mathematical induction is a technique for proving theorems. It goes with a method for defining functions called recursion. A typical definition by recursion is the following: Definition 4.1.2. We define n!, for all n ∈ N, as follows:

1. 0! = 1,
2. for all n, (n + 1)! = (n + 1). n!. So we define a function with domain N by saying what value it takes at 0, and saying how to get from the value it takes at n, to the value it takes at n + 1. The function defined above is the familiar factorial function, which we are used to defining informally thus: n! = 1. 2. 3..... n.

One use of the factorial function is to define the following extremely useful function:

Definition 4.1.3. If m and n are natural numbers, with m ≤ n, then we define ( n m

n! m!(n − m)!

We will be discussing the Principle of Mathematical Induction later; in particular, the difficulties involved in creating arguments by mathematical induction. But for now, let us leave the subject with an example of Mathematical Induction in action:

Theorem 4.1.4. For all natural numbers n, for all real numbers a and b,

(a + b)n^ =

∑^ n

k=

n k

akbn−k^.

In order to prove this I need a subordinate theorem to help me. In the trade, this is known as a lemma.

Lemma 4.1.5. Sppose that n and k are natural numbers, and 1 ≤ k ≤ n. Then (^) ( n + 1 k

n k

n k − 1

Proof: Exercise. 

Proof of Theorem: We prove this using induction on n. Base case: If n = 0, then (a + b)n^ = (a + b)^0 = 1; and ∑^ n

k=

n k

akbn−k^ =

a^0 b^0 = 1.

Inductive step: Now suppose the result true for n; we prove it for n + 1. We assume the inductive hypothesis: that

(a + b)n^ =

∑^ n

k=

n k

akbn−k^.

Then

(a + b)n+1^ = (a + b)n^. (a + b)

=

( (^) ∑n

k=

n k

akbn−k

. (a + b)

( (^) ∑n

k=

n k

akbn−k

.a +

( (^) ∑n

k=

n k

akbn−k

.b

( (^) ∑n

k=

n k

ak+1bn−k

( (^) ∑n

k=

n k

akbn−k+

(n∑+

k=

n k − 1

akbn+1−k

( (^) ∑n

k=

n k

akbn+1−k

n 0

a^0 bn+1^ +

∑^ n

k=

n k − 1

n k

akbn+1−k^ +

n n

an+1b^0.