Chapter 6: Section 6-1 Sets and Sample Spaces · PDF fileChapter 6: Section 6-1 Sets and...

Chapter 6: Section 6-1Sets and Sample Spaces

D. S. MalikCreighton University, Omaha, NE

D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-1 Sets and Sample Spaces 1 / 42

Sets

Cantor is considered to be the founder of set theory.

He explored paradoxes that had existed in mathematics for centuriesand even stumbled upon his own, known as Cantor’s paradox.Although his theories were vehemently disputed by his peers,including Kronecker, his mentor at University of Berlin, modernmathematicians completely accept Cantor’s work.

What is a set after all?

It is fascinating to know that the answer to this very obvious andtame looking question once put the very foundation of the theory ofsets into jeopardy.

However, in this text we adopt a naive and intuitive point of view andintroduce the definition of a set after Cantor:

“A set is a well-defined collection of distinct objects of our perception orof our thoughts, to be conceived as a whole.”


“A set is a well-defined collection of distinct objects of our perception orof our thoughts, to be conceived as a whole.”

Note that, here well-defined is an adjective to the collection and notto the distinct objects that are to be collected to form a set.

By this, it is meant that there should be no ambiguity whatsoeverregarding the membership of such a collection. In other words,well-defined means we can tell for certainty whether an object is amember of the collection or not.

These objects are called members or elements of the set.A collection of some positive integers is not a set, as it is not clearwhether a particular positive integer, say 5, is a member of thiscollection or not.

The collection of students taking a college algebra course in yourschool is a set.

The collection of best cars in a city will not be a set as there is nowell-defined notion of “best.”


We use uppercase letters A, B, C , . . . , X , Y , Z to denote sets.Two common ways to describe a set:

Roster methodSet builder method


Roster Method

Elements are separated with commas and enclosed within curly braces.

For example, if A is a set of vowels, then we write

A = {a, e, i , o, u}.

We can describe the set B of all positive integers less than 11 as

B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.


Let X be a set.

If x is an element of X , then we write x ∈ X and say that x belongs toX . The symbol ∈ stands for belongs to, which, like many othernotations, was introduced in 1889 by the Italian mathematician G.Peano (1858-1932), and which is believed to be a stylized form of theGreek epsilon.If x is not an element of X , then we write x /∈ X and say that x is notan element of X . The symbol /∈ stands for does not belong to.

ExampleLet A be the set

A = {1, 2, 3, 4, 5}.Then 2 ∈ A and 5 ∈ A. Also, 6 /∈ A.


Example

Consider the following collections of objects: 1, 2, 3, 4, {3, 4}, 5, {6, 7}.Each of the objects in this collection is either a number or a set ofnumbers. So this collection is well-defined and thus forms a set. Let uswrite

A = {1, 2, 3, 4, {3, 4}, 5, {6, 7}}.Then 1 ∈ A, 3 ∈ A, 4 ∈ A, {3, 4} ∈ A, {6, 7} ∈ A. Because 6 does notappear as an element in A, 6 /∈ A.


Set Builder Method

Let S be a set. The notation

A = {x | x ∈ S ,P(x)}

orA = {x ∈ S | P(x)}

means that A is the set of all elements x of S such that x satisfies theproperty P.

The vertical line | is read as “such that”.This way of describing a set is called the set-builder method.


ExampleLet

X = {0, 2, 4, 6}.In the set builder notation X can be written as

X = {x | x is an even integer and 0 ≤ x ≤ 6}.

Here the property P(x) is

P(x) : x is an even integer and 0 ≤ x ≤ 6.

Note that X can also be written as:

X = {x | x is a nonnegative even integer and x ≤ 6}


P(x) : x is a nonnegative even integer and x ≤ 6.


Example

Let A = {2,−2}. Because 2 and −2 are the only integers that satisfy theequation x2 − 4 = 0, we can also write A as

A = {x | x is an integer such that x2 − 4 = 0}.


P(x) : x is an integer such that x2 − 4 = 0


Subset

Let Z denotes the set of integers and R denotes the set of realnumbers.

Every element of Z is an element of R.

Let A = {a, e, i , o, u} and B is the set of all English letters.Then every element of A is an element of B.

When every element of a set, say A, is also an element of a set, sayB, we say that A is a subset of B.

DefinitionLet X and Y be sets. Then X is said to be a subset of Y , writtenX ⊆ Y , if every element of X is an element of Y . If X is not a subset ofY , then we write X * Y .


ExampleLet

X = {0, 2, 4, 6, 8},Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10},

andZ = {1, 2, 3, 4, 5}.

Then X ⊆ Y because every element of X is an element of Y .

Because 0 ∈ X and 0 /∈ Z , we have X * Z .Notice that we used the fact that 0 ∈ X and 0 /∈ Z to conclude thatX * Z .We could have also used the fact that 6 ∈ X and 6 /∈ Z or 8 ∈ X and8 /∈ Z , to conclude that X * Z .In other words, the elements 6 or 8 also prevent X from being asubset of Z .


Example

Let A = {a, b, c} and B = {a, c , b}. Element of A is also an element ofB, so A ⊆ B. Also note that B ⊆ A.

Example

Let A = {1, 2, 3, {3}, 4, 5, {4, 5}}.The elements of A are 1, 2, 3, {3}, 4, 5, and {4, 5}.Because 2 is an element of A, 2 ∈ A.Because 3 is an element of A, so 3 ∈ A.{3} appears as an element in the set A. So {3} ∈ A.3 is an element of A, so {3} ⊆ A.4 ∈ A, 5 ∈ A, and {4, 5} ∈ A.Because 4, 5 ∈ A, we have {4, 5} ⊆ A.


RemarkFor every set X , we have X ⊆ X , i.e., every set is a subset of itself.

DefinitionLet X and Y be sets. If X ⊆ Y , we also say that X is contained in Y orY contains X or Y is a superset of X written as, Y ⊇ X .

DefinitionLet X and Y be sets. Then X is a proper subset of Y , written X ⊂ Y , ifX is a subset of Y and there exists at least one element in Y which is notin X .

Example

Let A = {a, b} and B = {a, b, c}. Because every element of A is anelement of B, we have A ⊆ B. Now c ∈ B and c /∈ A. Therefore, thereexists an element in B that is not in A. It now follows that A is a propersubset of B, i.e., A ⊂ B.


DefinitionTwo sets X and Y are said to be equal, written X = Y , if every elementof X is an element of Y and every element of Y is an element of X , i.e., ifX ⊆ Y and Y ⊆ X .

Example

(i) {a, b, c} = {a, c , b}.(ii) Let A = {1, 2, 3, 4} and B = {x | x is a positive integer and x2 < 18}.Then A = B.(iii) The set A = {x | x is an integer and x3 = 1} and B = {1} are equal.


Empty Set

Let A = {x | x is an integer and x2 − 2 = 0}.Now x2 − 2 = (x −

√2)(x +

√2).

Thus, the solutions of the equation x2 − 2 = 0 are√2 and −

√2 and

none of these is an integer.Therefore, A does not contain any element.This is an empty collection of objects.We call it an empty set.

DefinitionA set is said to be an empty set (or, null set) if it has no elements. Wedenote an empty set by the symbol ∅.

RemarkThe empty set is a subset of every set. In fact, if ∅ * A for some set A,then there exists an element x ∈ ∅ such that x /∈ A. However, there is nosuch element x because ∅ is empty. Hence, ∅ ⊆ A for every set A.


Example

Let A = {a, e, i , o, u}. Then A is a set with 5 elements. Such a set iscalled a finite set.

DefinitionLet X be a set.(i) If there exists a nonnegative integer k such that X has k elements thenX is called a finite set with k elements.(ii) X is called an infinite set, if X is not a finite set.


Example

(i) The set A = {a, b, c} has three elements, so it is a finite set.(ii) The set B of the first 10 positive odd integers is a finite set. Note that

B = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}.

(iii) The set of positive integers is an infinite set.

RemarkNote that an empty set is a finite set with 0 elements.

Notation: Let S be a finite set with k distinct elements, where k ≥ 0.Then we write n(S) = k and say that the number ofelements in S is k.


Example

If A = {a, b, c , d , e}, then A is a finite set with five elements, so n(A) = 5.

ExampleLet

B = {x | x is an even prime integer}.The only even prime integer is 2. Therefore, B = {2}, so n(B) = 1.


Let X = {1, 2}. Then

∅ ⊆ X , {1} ⊆ X , {2} ⊆ X , and X ⊆ X .

Now each of the sets ∅, {1}, {2}, X is well defined. We cantherefore form the collection, {∅, {1}, {2},X}, of these sets, whichwould itself be a set.

DefinitionFor any set X , the power set of X , written P(X ), is the set of all subsetsof X . That is,

P(X ) = {A | A ⊆ X}.

Example

For example, let X = {a, b, c}. Then

P(X ) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c},X}.

Notice that n(X ) = 3 and n(P(X )) = 8 = 23.D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-1 Sets and Sample Spaces 20 / 42

Theorem

Let X be a finite set X such that n(X ) = k. Then n(P(X )) = 2k .

RemarkBecause ∅ is a subset of every set, we have ∅ ⊆ ∅. Therefore, we haveP(∅) = {∅}. Note that n(∅) = 0 and n(P(∅)) = 20 = 1.


Universal Set

To avoid the logical diffi culties that arise in the foundation of set theory,we further assume that each discussion involving a number of sets takesplace with respect to an arbitrarily chosen but fixed set. This set is calleda universal set for that discussion and is generally denoted by U. It has tobe a set such that all the sets under consideration in that problem aresubsets of U.

RemarkIt may clearly be understood– though the name may sound to suggestotherwise– that by no means are we proposing a set which is universal forall the problems, rather it may vary from problem to problem and evenmore– for a problem involving certain sets, the choice of a universal set isnot unique, but once chosen, subject to the conditions stated above, itmust be kept fixed throughout the subsequent discussions of that problem.


Example

In a discussion involving the sets X = {1, 2, 3}, Y = {2, 4, 6, 8}, andZ = {1, 3, 5, 7} one may choose U = {1, 2, 3, 4, 5, 6, 7, 8} as a universalset. Moreover any superset of U can also be considered a universal set forthese sets X , Y , and Z . For example, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} canalso be considered a universal set for the sets X ,Y , and Z .


Venn Diagrams

Typically, it is not easy to visualize a set. However, in 1880, the Englishlogician John Venn (1834-1923) gave a pictorial representation for setsand their fundamental operations. These are called Venn diagrams,where the universal set U is represented by a rectangle and all its subsetsunder consideration by circles drawn within the rectangle, as shown in thefollowing figure. (The shaded portion in a Venn diagram represents thecorresponding set.)

XU


Operations on Sets

Given two sets A and B, there are various ways we can form new setsusing the sets A and B.

For example,

We can form the set by taking all elements from A and all elementsfrom B, (in this case, an element will not be considered more thanonce).We can also form the set by taking elements that are common to boththe sets A and B.We can form the set by taking all of the elements of A that are not inB.


Union Sets

Definition

The union of two sets X and Y , denoted by X ∪ Y , is defined to be theset

X ∪ Y = {x | x ∈ X or x ∈ Y }.

Note that in Definition 24, x ∈ X or x ∈ Y means x is an element of Xor x is an element of Y or x is an element of both X and Y . In otherwords, x ∈ X ∪ Y if x is a member of at least one of the sets X and Y .The Venn diagram of the union of sets is:

X

U

Y


Example

Let X = {a, b, c , d , e}, Y = {c, d , e, f , g , h}, and Z = {h, p, q, r}. Then

X ∪ Y = {a, b, c, d , e, f , g , h},

X ∪ Z = {a, b, c, d , e, h, p, q, r},and

Y ∪ Z = {c, d , e, f , g , h, p, q, r}.

TheoremLet X and Y be sets. Then X ⊆ X ∪ Y and Y ⊆ X ∪ Y .


Intersection of Sets

DefinitionThe intersection of two sets X and Y , denoted by X ∩ Y , is defined tobe the set

X ∩ Y = {x | x ∈ X and x ∈ Y }.

The Venn diagram of the intersection of sets is:

X

U

Y


Example

Let X = {a, b, c , d , e}, Y = {c, d , e, f , g , h}, and Z = {h, p, q, r}. Then

X ∩ Y = {c, d , e},

X ∩ Z = ∅,

andY ∩ Z = {h}.

TheoremLet X and Y be sets. Then X ∩ Y ⊆ X and X ∩ Y ⊆ Y .


Disjoint Sets

DefinitionTwo sets X and Y are said to be disjoint if X ∩ Y = ∅.

It is not possible to draw the Venn diagram of the null set by shading;however disjoint sets are represented as follows:

X

U

Y

Example

Let U = {Math, English, History , Chemistry , Spanish, Physics,Sociology}. Let A = {Math, Spanish, Physics} and B = {English,History , Sociology}. Then A∩ B = ∅. Thus, A and B are disjoint.


Venn Diagrams of Three Sets

Let X , Y , and Y be subsets of a set U. The Venn diagram of X ∪ Y ∪ Zand X ∩ Y ∩ Z are:

X

U

Y

Z

X

U

Y

ZX ∪ Y ∪ Z X ∩ Y ∩ Z


Mutually Disjoint or Pairwise Disjoint Sets

DefinitionLet A = {Aα | α ∈ I} be a family of sets. The sets Aα are said to bemutually disjoint or pairwise disjoint if for α, β ∈ I , α 6= β impliesAα ∩ Aβ = ∅.

Example

Let A1 = {a, b, c}, A2 = {d , e}, and A3 = {u, v ,w}. Then A1 ∩ A2 = ∅,A2 ∩ A3 = ∅, and A1 ∩ A3 = ∅. This implies that the sets A1,A2, and A3are pairwise disjoint.


Difference or Relative Complement of Sets

DefinitionLet X and Y be sets. The difference of X and Y (or the relativecomplement of Y in X ), written X − Y is the set,

X − Y = {x | x ∈ X but x 6∈ Y }.

The Venn diagram of the difference of sets is:

X

U

Y

Example

Let X = {1, 2, 3, 4} and Y = {3, 4, 5, 6}. Then X − Y = {1, 2} andY − X = {5, 6}. Note that X − Y 6= Y − X , i.e., the difference of sets isnon-commutative.


Complement of a Set

DefinitionThe complement of a set X with respect to a universal set U, denoted byX ′ is defined to be

X ′ = {x ∈ U | x 6∈ X}.

From the preceding definition, it follows that X ′ = U − X .The Venn diagram of the complement of a sets is:

XX' U


Example

(i) Let U = {a, b, c, d , e, f , g , h} and A = {a, c , g , h}. Then

A′ = {b, d , e, f }.

(ii) Let Z be the set of integers and E be the set of even integers. ThenE′ = Z−E = {x ∈ Z | x is an odd integer}.


ExampleLet X ,Y ,Z be subsets of a set U. The Venn diagrams of the setsX − (Y ∪ Z ) and (X ∪ Y )− Z is:

X

U

Y

Z

X

U

Y

ZX − (Y ∪ Z ) (X ∪ Y )− Z


TheoremLet X and Y be sets and U be a universal set under consideration. Then(i) X ∪ X ′ = U and X ∩ X ′ = ∅(ii) (X ′)′ = X(iii) X − Y = X ∩ Y ′(iv) (De Morgan’s laws) (X ∪ Y )′ = X ′ ∩ Y ′ and(X ∩ Y )′ = X ′ ∪ Y ′.


Using Venn diagrams, we verify that X − Y = X ∩ Y ′.

X

U

Y

(a) X − Y

X

U

Y X

U

Y

(b) X (c) Y ′

X

U

Y X

U

Y

(d) X and Y ′ (e) X ∩ Y ′D. S. Malik Creighton University, Omaha, NE ()Chapter 6: Section 6-1 Sets and Sample Spaces 38 / 42

Ordered Pair and Cartesian Cross Product

Let X and Y be sets. Intuitively, an ordered pair of the elements x ∈ Xand y ∈ Y , written (x , y), is a listing of the elements x and y in a specificorder. The ordered pair (x , y) specifies that x is the first element and y isthe second element. Moreover, we use the convention that (x , y) = (z ,w)if and only if x = z and y = w , for all x , z ∈ X and y ,w ∈ Y .

DefinitionThe Cartesian product of two sets X and Y , written X × Y , is the set

X × Y = {(x , y) | x ∈ X , y ∈ Y }.

For any set X , X ×∅ = ∅ = ∅× X .


Example

Let X = {1, 2}, Y = {3, 4}. Then

X × Y = {(1, 3), (1, 4), (2, 3), (2, 4)}

andY × X = {(3, 1), (3, 2), (4, 1), (4, 2)}.

Suppose that X = {a, b, c} and Y = {1, 2}. Then

X × Y = {(a, 1), (a, 2), (b, 1), (b, 2), (c , 1), (c, 2)}.

Thus, we have n(X ) = 3, n(Y ) = 2, and n(X × Y ) = 3 · 2 = 6.

TheoremLet X and Y be sets such that n(X ) = m and n(Y ) = t. Thenn(X × Y ) = mt.


Exercise: Let A and B be subsets of the set U. Draw the Venndiagrams of A′ ∪ B and (A∩ B)′.

Solution:

A

U

B A

U

B

A′ ∪ B (A∩ B)′


Exercise: Let A,B, and C be subsets of the set U. Draw the Venndiagrams of (A∩ B ′) ∪ C and (A∩ B)′ ∩ C .

Solution:

A

U

B

C

A

U

B

C

(A∩ B ′) ∪ C (A∩ B)′ ∩ C


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