Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed...

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Mathematical Proofs

Transcript of Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed...

Page 1: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Mathematical Proofs

Page 2: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Chapter 1 Sets

• 1.1 Describing a Set• 1.2 Subsets• 1.3 Set Operations• 1.4 Indexed Collections of Sets• 1.5 Partitions of Sets

Page 3: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Section 1.1 Describing a Set

A set is a collection of objects. The objects that make up the set are called its elements.

It’s customary to use capital (uppercase) letters (such as A, B, C, S, X, Y) to describe sets and lowercase letters (for example, a, b, c, s, x, y) to represent elements of sets.

If a is an element of the set A, then we write aA; if a does not belong to A, then we write a A.

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List All the Elements

If a set consists of a smaller number of elements, then it can be described by explicitly listing its elements between braces where the elements are separated by commas.

Example: S = {1, 2, 3} is a set.

Note that the order in which the elements are listed doesn’t matter.

Example: S = {1, 2, 3} = {1, 3, 2} = {2,1,3} etc.

If a set contains too many elements to be listed, then we use the ellipsis or “three dot notation”.

Example: The set of all positive even integers less than 41 can be described by X={2, 4, …, 40}

The set of all positive odd integers can be described by Y={1, 3, 5, …}

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Empty Set

There is one set that contains no elements, and it is called the empty set, denoted by . We also write ={}.

Example: The set of real number solutions of equation x2+1=0, then S=.

Note: The elements of a set may in fact be sets themselves.

Example: The set S={a, b, {c, d}, e, } has 5 elements.

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Describe the Property

Often sets consist of those elements satisfying some specified property. In this case, we can describe it as S={x: p(x)} , which means S consists of all those elements x satisfying some condition p(x) concerning x.

Example: T={x: |x|=3}.

For a set S, the cardinality of S is the number of elements in S, denoted by |S|.

Example: if S={a, b, {c, d}, e, }, then |S|=5. Also, | |=0.

A set S is finite if |S|=n for some nonnegative integer n. A set S is infinite if it is not finite.

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Special Sets

Some sets are given special notations.

N: the set of natural numbers. (positive integers.) N={1, 2, 3, …}

Z: the set of all integers (positive, negative, and zero).

Z={…,-1, 0, 1, …}

Q: the set of rational numbers. (m/n, where m, n Z and n 0).

I: the set of all irrational numbers. (a real number that is not rational.)

R: the set of real numbers

R+: the set of positive real numbers.

C: the set of complex numbers. ( a number of the form a+bi, where a, b R and )1i

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Section 1.2 Subsets

A set A is called a subset of a set B is every element of A also belongs to B. If A is a subset of B, then we write A B. If a set C is not a subset of D, then we write CD. That is, there must be some element of C that is not an element of D.

Note that the empty set is a subset of every set.

If A, B, and C are sets such that A B and B C, then A C. Moreover, every set is a subset of itself.

For example: N Z, Q R, and R C.

Let S={1, {2}, {1, 2}}. (a) Determine which are elements of S, (b) Determine which are

subsets of S.

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More Definitions

Two sets A and B are equal, A=B, if they have exactly the same elements. Equivalently, A=B happens if and only if A B and BA.

A set A is a proper subset of a set B if A B but AB. If A is a proper subset of B, then we write AB. The set consisting of all subsets of a given set A is called the power set of A and it denoted by P(A).

Example: For each set A below, determine P(A), |A| and |P(A)|.(a) A= , (b) B={1, 2}, (c) C={a, b, c}, (d) D={, {}}.

Note that |P(A)|=2|A|.

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Venn Diagrams

This is an example of Venn diagram:

x yz

w

AB

This presents two sets A and B. A rectangle in a Venn diagram represents the universal set (the set we are considering).

Note xA, xB; yB, yA; z A, z B; w A, w B.

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Interval

Some frequently encountered subsets of R are the so-called “intervals”.• For a, b R and a<b, the open interval (a, b) is the set (a, b)={x R: a < x < b}.• For a, b R and a b, the closed interval [a, b] is the set [a, b]={x R: a x b}.• For a, b R and a b, the half-open interval [a, b), (a, b] are the

sets [a, b)={x R: a x < b}, and (a, b]={x R: a < x b}.

For a R, the infinite intervals (-, a), (-, a], (a, ) and [a, ) are defined as

(-, a)={x R: x < a}, (-, a]={x R: x a}, (a, )={x R: x >a}, [a, )={x R: x a}.

Note that (-, ) is the set R.

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Section 1. 3 Set Operations

The union of two sets A and B, denoted by AB, is the set of all elements belonging to A or B. That is,

AB={x: x A or x B}.

A Venn diagram for AB is

Example: Let A={a, b}, B={a, c, d} and C={b, e, g}. Find AB, AC and BC.

Note: N Z=Z and Q I=R.

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Intersection

The intersection of two sets A and B, denoted by A B, is the set of all elements belonging to both A and B. That is,

A B={x: x A and x B}.

A Venn diagram for A B is

Example: Let A={a, b}, B={a, c, d} and C={b, e, g}. Find A B, A C and B C.

Note: N Z=N and Q I= .If two sets A and B have no elements in common, then A B= and

A and B are said to be disjoint.

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Difference

The difference A-B of two sets A and B is defined as

A - B={x: x A and x B}.

A Venn diagram for A - B is

Example: Let A={a, b}, B={a, c, d} and C={b, e, g}. Find A - B, C - B and A C.

Note: R-Q=I.

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Complement

Suppose that we are considering a certain universal set U. For a set A, its complement is

A Venn diagram for is

{ }.A U A x U and x A

A

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Examples

Example: Let U={1, 2, …, 10} be the universal set, A={1, 2, 4, 8} and B={1, 3, 5, 7, 9}. Determine each of the following:

(a) , (b) B- A, (c) A , (d)

Example: Let A={0, {0}, {0, {0}}}.

(a) List all the elements of A,

(b) List all the subsets of A,

(c) Determine |A|,

(d) {0} A, {{0}} A, {{{0}}} A, {0} A, {{0}} A, {{{0}}} A.

B B B

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Section 1.4 Indexed Collections of Sets

We will often encounter situations where more than two sets are combined using the set operations described in section 1.3.

The union A B C is defined as

A B C={x: x A or x B or x C}

Generally, the union of the n2 sets A1, A2,…,An is denoted by

A1A2…An={x: x Ai for some i, 1in}

Thus, for an element a to belong to , it is necessary that a belongs to at least one of the sets A1, A2,…,An.

1ni iA

Page 18: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Examples

Example: Let Bi={I, i+1} for i=1, 2, …, 10. Determine each of the following:

Similarly, the intersection of the n2 sets A1, A2,…,An is denoted by

A1 A2 … An, or and is defined by

={x: x Ai for every i, 1in}

Example: Let Bi={I, i+1} for i=1, 2, …, 10. Determine each of the following:

1ni iA

5 8

1 2

) , ) , ) , where 1 8.k

i i ii i i j

a B b B c B j k

1ni iA

10

11

) , ) , ) , where 1 8.k

i i i ii i j

a B b B B c B j k

Page 19: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Indexed Collection of Sets

We introduce a (nonempty) set I, called an index set. For an index set I, suppose that there is a set S for each I.

We write {S } I to describe the collection of all sets S , where I. Such a collection is called an indexed collection of sets.

We define the union of the sets in {S } I by

An element a belongs to if a belongs to at least one of the sets in the collection {S } I

We refer to as the union of the collection {S } I I

S

{ : for some }I

S x x S I

I

S

Page 20: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Intersection

We define the intersection of the sets in {S } I by

An element a belongs to if a belongs to every set in the collection {S } I

We refer to as the intersection of the collection {S } I.

Here the variables I and are “dummy variables” and any appropriate symbol could be used.

{ : for all }I

S x x S I

I

S

I

S

Page 21: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Example

Example: For nN, define Sn={n, 2n}. Then S1S2 S3={1, 2, 4, 8}. If we let I={1, 2, 4}, then

Example: For each nN, define An to be the closed interval [-1/n, 1/n] of real numbers. That is, An={xR: -1/nx 1/n}.

Then n N An=[-1, 1]. In fact, n N An=0.

1 2 3I

S S S S

Page 22: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Section 1.5 Partitions of Sets

Recall that two sets are disjoint if their intersection is the empty set. A collection S of subsets of a set A is called pairwise disjoint if every two distinct subsets that belong to S are disjoint.

A partition of A can be defined as a collection S of nonempty subsets of A such that every element of A belongs to exactly one subset in S.

Furthermore, a partition of A can be defined as a collection S of subsets of A satisfying the three properties:

1. X for every set XS;

2. For every two sets X, Y S, either X=Y or XY=;

3. X SX=A.

Page 23: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Example

Example: Let A={1, 2, 3, 4, 5, 6}. Determine which are the partitions of A.

1. {{1, 3, 6}, {2, 4}, {5}};

2. {{1, 2, 3}, {4}, , {5, 6}};

3. {{1, 2}, {3, 4, 5}, {5, 6}};

4. {{1, 4}, {3, 5}, {2}};

Example:

1. Z can be partitioned into the set of even integers and the set of odd integers.

2. R can be partitioned into R+, the set of negative real numbers, and {0}. It can also be partitioned into the set Q and the set I.

Page 24: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Section 1.6 Cartesian Product of Sets

The ordered pair (x, y) is a single element consisting of a pair of elements in which x is the first coordinate of the ordered pair (x, y) and y is the second coordinate.

If two ordered pairs (x, y) and (w, z) are equal, that is (x, y)=(w, z), then x=w and y=z. So if x y, then (x, y) (y, x).

The Cartesian product A X B of two sets A and B is the set consisting of all ordered pairs whose first coordinate belongs to A and whose second coordinate belongs to B. That is,

A X B = {(a, b): a A and b B}.

Page 25: Mathematical Proofs. Chapter 1 Sets 1.1 Describing a Set 1.2 Subsets 1.3 Set Operations 1.4 Indexed Collections of Sets 1.5 Partitions of Sets.

Example

Example: If A={x, y, z} and B={1, 2}, then find AXB, BXA, AXA, and BXB.

Note if A= or B= , then AXB= .

Also, for all finite sets A and B, |AXB|=|A|X|B|.