Download Sets, Functions and Mathematical Induction

Chapter 1
Introductory Material:
Sets, Functions and
Mathematical Induction
1.1
Review of Sets
This section is only intended as a quick review of material the reader needs
to know for these notes. For a more thorough treatment of sets, the reader is
invited to read books on set theory.
De…nition 1 (Set) A set is a well-de…ned collection of distinct objects. Wellde…ned means that given an object and a set, we can tell if the object is in the
set. The objects in a set are called its elements. If x is an element of a set A,
we write x 2 A. This is read "x is an element of A".
A set can be …nite or in…nite. Sets are usually named using a capital letter
such as A, B, ... The words sets, collection, family are all synonymous. The
order in which the elements are listed is not relevant. In other words, two sets
are equal if and only if they contain the same elements, regardless of the order
in which they appear.
1.1.1
Examples and Notation
There are di¤erent ways of representing a set.
1. We can list all the elements of the set, or, we list enough elements until a
pattern is established, then we add ”...”. In both cases, the elements are
surrounded by curly brackets.
Example 2 A = f1; 2; 4g represents the set consisting of the numbers 1,
2, and 4.
1
2CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDUC
Example 3 B = f1; 2; 3; :::g represents the set of natural numbers.
Example 4 P = ff1; 2g ; f2; 3gg. The elements of a set can be sets themselves.
2. We give a rule or a condition to be satis…ed in order to belong to a set. If
P (x) denotes the condition to be satis…ed, we can de…ne a set by writing
fx : P (x)g. This means the set of x0 s such that P (x) is true. Since this
is real analysis, we will assume that the x0 s come from R. If this is not
the case, we must specify the set they come from. If they come from a set
A, then we write fx 2 A : P (x)g.
Example 5 A = fx 2 N : x is eveng is the same as f2; 4; 6; :::g
Remark 6 In the above de…nition, : is read "such that".
Example 7 C = fx 2 R :
Example 8 Q=
1.1.2
nm
n
2 < x < 5g
o
: m 2 Z, n 2 Z and n 6= 0
Special Symbols
There are some special symbols associated with sets. Some of these symbols
include:
2 means ”is a member of”. When we write 2 2 A, we mean that the
element 2 is a member of the set called A.
2
= means ”is not a member of”.
means "is included in but not equal to" or ”is a proper subset of”.
It is used between two sets. When we write A
B, we mean that A is
contained in B, the two are not equal.
means "is included in and could be equal to" or ”is a subset of”; it is
used between two sets. When we write A B, it is understood that A is
contained in B and could be equal to B:
? denotes the empty set. It is also called the void set or the null set.
If A is a set, we denote jAj the cardinality of A that is the number of
elements of A.
Remark 9 Let A and B be two sets.
1. To show A B, one must show that if we pick an arbitrary element x in
A then x is also in B.
1.1. REVIEW OF SETS
3
2. To show A B one must show that A
element which is not in A.
3. It should be clear that ?
B and that B has at least one
A for every set A.
Some sets which are used often have a special name.
N = f1; 2; 3; :::g represents the set of natural numbers or positive integers.
Z = f:::; 2; 1; 0; 1; 2; :::g represents the set of integers.
Q=
nm
n
o
: m; n 2 Z and n 6= 0 represents the set of rational numbers.
R represents the set of real numbers.
If A is any set, then the power set of A, denoted P (A), is the set of all
subsets of A (including ? and A). For example if A = f1; 2; 3g then
P (A) = f?; f1g ; f2g ; f3g ; f1; 2g ; f1; 3g ; f2; 3g ; f1; 2; 3gg
It can be shown that if A has n elements, then P (A) has 2n elements that
is jP (A)j = 2n .
1.1.3
Operations on Sets
There are four operations on sets you should know about. They are [ (union),
\ (intersection), (or n) (di¤erence or relative complement) and (Cartesian
product). These operations are de…ned below. Let us assume we have two sets
A and B.
Union
The union of A and B, denoted A [ B is de…ned by:
A[B = fx : x 2 A or x 2 Bg i.e. it consists of the elements which are either
in A or in B.
Intersection
The intersection of A and B, denoted A \ B is de…ned by:
A \ B = fx : x 2 A and x 2 Bg i.e. it consists of the elements which are in
both A and B.
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDUC
Figure 1.1: Union of Sets: A [ B
Relative Complement
The relative complement of B in A, denoted A n B or A B (we will use A n B
as A B also means fa b : a 2 A and b 2 Bg) is de…ned by:
A n B = fx : x 2 A and x 2
= Bg i.e. it consists of the elements of A which
are not also in B.
Sometimes, when all the sets we are working with are subsets of a …xed set
we will call X, then instead of writing X n A or X A, we write AC . This set
is called the complement of A. It is understood that it is the complement of
A in X. For example, in these notes, R is the set where all our elements come
from, unless speci…ed otherwise. So, if we write AC , it will mean R A.
If A and B are two sets, A n B 6= B n A in general.
Cartesian Product
The Cartesian product of A and B, denoted A B is de…ned by:
A B = f(x; y) : x 2 A and y 2 Bg. This is called the Cartesian product of
A and B. It is the set of all ordered pairs whose …rst element comes from A
and second element from B.
Because it is a set of ordered pairs, if A and B are di¤erent sets, A B 6=
B A.
Remark 10 The Cartesian product of R with itself, R
R is denoted R2 .
Figures 1.1, 1.2 and 1.3 illustrate the …rst three operations.
1.1. REVIEW OF SETS
5
Figure 1.2: Intersection of Sets: A \ B
Figure 1.3: Di¤erence of Sets: A
B
6CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDUC
Example 11 Given A = f0; 1; 2; 3; 4; 5; 6g and B = f5; 6; 7; 8; 9g, then we have:
A [ B = f0; 1; 2; 3; 4; 5; 6; 7; 8; 9g
A \ B = f5; 6g
A n B = f0; 1; 2; 3; 4g
B n A = f7; 8; 9g
Example 12 Given A = fa; bg and B = f1; 2g we have:
A
B
B = f(a; 1) ; (a; 2) ; (b; 1) ; (b; 2)g
A = f(1; a) ; (1; b) ; (2; a) ; (2; b)g
Remark 13 From the two examples above, you notice that, in general, A n B 6=
B n A and A B 6= B A.
1.1.4
Some Results About Sets
Let us make a few remarks before we state and prove some important results.
Remark 14 The standard technique to prove that two sets A and B are equal
(that is A = B) is to prove that A B and B A.
Remark 15 Often, when proving some result about the elements of a set A,
we start by picking an arbitrary element of that set by using a statement like
"Let x 2 A....". However, this statement makes sense only if A has elements,
in other words A 6= ?. If we do not know that fact, we will have to consider the
special case A = ? as part of the proof.
Theorem 16 Let A; B; and C be three sets. Then, the following properties are
satis…ed:
1. A \ A = A, A [ A = A
2. A \ ? = ?, A [ ? = A
3. Commutative law. A [ B = B [ A and A \ B = B \ A
4. Associative law. A[(B [ C) = (A [ B)[C and A\(B \ C) = (A \ B)\C
5. Distributive laws. A [ (B \ C) = (A [ B) \ (A [ C) and A \ (B [ C) =
(A \ B) [ (A \ C)
6. If in addition, A
B then A [ B = B and A \ B = A
Proof. See problems at the end of the section.
De…nition 17 (Disjoint Sets) Two nonempty sets A and B are said to be
disjoint if they do not intersect that is if A \ B = ?.
1.1. REVIEW OF SETS
7
Remark 18 It should be clear to the reader that if A = ? or B = ? then
A \ B = ?.
Theorem 19 A \ B and A n B are disjoint sets. Furthermore, A = (A \ B) [
(A n B).
Proof. We …rst prove the sets are disjoint. We then prove A = (A \ B) [
(A n B), by showing that A (A \ B) [ (A n B) and (A \ B) [ (A n B) A.
Proof that the sets are disjoint. The result seems obvious if we look at
Figure 1.3. However, this is just a speci…c case. We need to establish this
fact in general. We show the sets are disjoint by showing an element x
cannot be in both sets at the same time. If x 2 A \ B, then it is in both A
and B. Hence, it cannot be in A n B. Similarly, if x 2 A n B, then x 2
=B
hence it cannot be in A \ B.
Proof that A
(A \ B) [ (A n B). If A = ?, the result follows since ?
is a subset of any set. If A 6= ?, the we pick x 2 A. Either x 2 B in
which case x 2 A \ B, or x 2
= B in which case x 2 A n B. It follows that
x 2 (A \ B) [ (A n B) hence A (A \ B) [ (A n B).
Proof that (A \ B) [ (A n B)
A. If (A \ B) [ (A n B) = ?, the result
follows since ? is a subset of any set. If (A \ B) [ (A n B) 6= ? then let
x 2 (A \ B) [ (A n B). Either x 2 A \ B or x 2 A n B. In both cases.
x 2 A hence (A \ B) [ (A n B) A.
Theorem 20 (De Morgan’s Laws) Let A, B, and C be sets. Then the following holds:
1. A n (B \ C) = (A n B) [ (A n C)
2. A n (B [ C) = (A n B) \ (A n C)
Proof. We only prove the …rst part. The second one is left as an exercise.
First, we show that A n (B \ C)
(A n B) [ (A n C). Let x 2 A n (B \ C).
Then, x 2 A and x 2
= B \ C. So, x 2 A and either x 2
= B or x 2
= C . If x 2
=B
then x 2 A n B since x 2 A. Similarly, if x 2
= C then x 2 A n C. Therefore,
either x 2 A n B or x 2 A n C. Hence, x 2 (A n B) [ (A n C).
Next, we show that (A n B) [ (A n C) A n (B \ C). Let x 2 (A n B) [ (A n C).
Either x 2 A n B or x 2 A n C. So, either x 2 A and x 2
= B or x 2 A and x 2
= C.
In both cases, x 2 A and either x 2
= B or x 2
= C. So, x 2 A and x 2
= B \ C that
is x 2 A n (B \ C).
Remark 21 If we assume that both A and B are subsets of a …xed set X and
we wish to express the relative complement with respect to X, then theorem 20
can be written
c
1. (A \ B) = Ac [ B c
c
2. (A [ B) = Ac \ B c
8CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDUC
1.1.5
Exercises
1. Prove Theorem 16.
2. Prove the second part of Theorem 20.
3. Prove that A
B () A \ B = A
4. Prove that A
B () A [ B = B
5. Let A and B be two non-empty sets.
(a) Prove that A \ B
(b) Prove that A
A
A[B
6. Let D denote the set of elements which either belong to A or to B but
not to both. Prove that D = (A n B) [ (B n A). This set is often called
the symmetric di¤erence of A and B.
7. Show that the set D de…ned in the previous problem is also given by
D = (A [ B) n (A \ B)
8. Read about Richard Dedekind and his contribution to the construction of
the set of real numbers.
9. Do the same for Georg Cantor.
10. Let
A
B
C
(a) Find the following:
A[B
B[C
A\B
B\C
A\C
A \ (B [ C)
AnB
C nB
A n (B [ C)
(b) Find the following:
A
C
B
B
= f 1; 0; 1; 2g
= f 2; 3g
=
f 2; 0; 1; 5g
1.1. REVIEW OF SETS
9
(A B) \ (C
(A \ C) B
B)
(c) On the basis of your answer in (b), what might you conjecture about
(A \ C) B for arbitrary sets A; B; C?
11. Let
A
B
C
= fx 2 R :
1
= fx 2 R :
2
= fx 2 R : 0
x
5g
x
3g
x
4g
(a) Find each of the following:
A\B
A\Z
B\C
A[B
B[C
(b) Find each of the following:
A B
A C
(A B) [ (A
A (B [ C)
(c) Sketch the sets (A
C)
B) [ (A
C) and A
(B [ C)
(d) On the basis of your answer in (c), what might you conjecture about
A (B [ C) for arbitrary sets A; B; C?
12. If A is a subset of X, prove that:
(a) A [ Ac = X
(b) A \ Ac = ?
c
(c) (Ac ) = A
13. Verify that a set with three elements has eight subsets.
14. If A and B are subsets of a set X, prove that A n B = A \ B c
15. If A and B are any sets prove that A n B and A \ B are disjoint and that
A = (A n B) [ (A \ B)
16. Prove that A (B1 [ B2 ) = (A
for intersection.
B1 ) [ (A
B2 ). Prove the same result
10CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU
17. Suppose that A; C are subsets of X and B; D are subsets of Y . Prove that
(A
B) \ (C
D) = (A \ C)
(B \ D)
18. What is jP (?)j?
19. Let A and B be sets. Prove that A \ B
A [ B.
20. Let A, E and F be three sets.
(a) Prove that if both E
A and F
A then E \ F
(b) Prove hat if A
E and A
F then A
(c) Prove hat if A
E and A
F then A
(d) Prove that if both E
(e) Prove that if E
(f) Prove that if E
A and F
F then A \ E
F then A [ E
A.
E [ F.
E \ F.
A then E [ F
A.
A \ F.
A [ F.
21. Let A, B and C be sets. Determine if the statements below are true or
false. If you think they are true, prove them. If you think they are wrong,
either explain why or give a counterexample.
(a) If A
(b) If A
B \ C then A
B [ C then A
B and A
B or A
C.
C.