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Basic Concepts of Set Theory: Symbols & Terminology
A set is a collection of objects.
A well-defined set has no ambiguity as to what objects are in the set or not.
For example: The collection of all red cars The collection of positive numbers The collection of people born before 1980 The collection of greatest baseball players
All of these collections are sets. However, the collection of greatest baseball players is not well-defined.
Normally we restrict our attention to just well-defined sets.
Defining Sets
word description The set of odd counting numbers between 2 and 12
the listing method { 3 , 5 , 7 , 9 , 11 }
set-builder notation or defining property method {x | x is a counting number, x is odd, and x < 12 }
Note: Use curly braces to designate sets, Use commas to separate set elements The variable in the set–builder notation doesn’t have to be x. Use ellipses (... ) to indicate a continuation of a pattern established before the ellipses i.e. { 1 , 2 , 3 , 4 ,... , 100 } The symbol | is read as “such that”
Set Membership
An element or member of a set is an object that belongs to the set
The symbol ∈ means “is an element of”
The symbol ∈/ means “is not an element of”
Generally capital letters are used to represent sets and lowercase letters are used for other objects i.e. S = { 2 , 3 , 5 , 7 }
Thus, a ∈ S means a is an element of S
Is 2 ∈ { 0 , 2 , 4 , 6 }? Is 2 ∈ { 1 , 3 , 5 , 7 , 9 }?
Some Important Sets
N — Natural or Counting numbers: {1, 2, 3,... }
W — Whole Numbers: {0, 1, 2, 3,... }
I — Integers: {... , -3, -2, -1, 0, 1, 2, 3,... }
Q — Rational numbers: { pq | p, q ∈ I, q 6 = 0 }
< — Real Numbers: { x | x is a number that can be written as a decimal }
Irrational numbers: { x | x is a real number and x cannot be written as a quotient of integers }. Examples are: π,
2 , and 3
∅ — Empty Set: { }, the set that contains nothing
U — Universal Set: the set of all objects currently under discussion
Notes
Any rational number can be written as either a terminating decimal (like 0.5, 0.333, or 0.8578966) or a repeating decimal (like 0. 333 or 123. 392545 )
The decimal representation of an irrational number never terminates and never repeats
The set { ∅ } is not empty, but is a set which contains the empty set
More Membership Questions
Is ∅ ∈ { a, b, c }?
Is ∅ ∈ { ∅, { ∅ } }?
Is ∅ ∈ { { ∅ } }?
Is 13 ∈ {/ x | x = (^1) p , p ∈ N}
Set Cardinality
The cardinality of a set is the number of distinct elements in the set
The cardinality of a set A is denoted n(A) or |A|
If the cardinality of a set is a particular whole number, we call that set a finite set
If a set is so large that there is no such number, it is called an infinite set (there is a precise definition of infinity but that is beyond the scope of this course)
Note: Sets do not care about the order or how many times an object is included. Thus, { 1 , 2 , 3 , 4 }, { 2 , 3 , 1 , 4 }, and { 1 , 2 , 2 , 3 , 3 , 3 , 4 , 4 } all describe the same set.
Set Cardinality
A = { 3 , 5 , 7 , 9 , 11 }, B = { 2 , 4 , 6 ,... , 100 }, C = { 1 , 3 , 5 , 7 ,... }
D = { 1 , 2 , 3 , 2 , 1 }, E = {x | x is odd, and x < 12 }
n(A) =? n(B) =? n(C) =? n(D) =? n(E) =?
Set Equality
Set Equality: the sets A and B are equal (written A = B) provided:
- every element of A is an element of B, and
- every element of B is an element of A
In other words, if and only if they contain exactly the same elements
{ a, b, c } = { b, c, a } = { a, b, a, b, c }? { 3 } = { x | x ∈ N and 1 < x < 5 }? { x | x ∈ N and x < 0 } = { y | y ∈ Q and y is irrational}?
Venn Diagrams & Subsets
Universe of Discourse – the set containing all elements under discussion for a particular problem In mathematics, this is called the universal set and is denoted by U Venn diagrams can be used to represent sets and their relationships to each other
A
U
Venn Diagrams
The “Universe” is represented by the rectangle
Sets are represented with circles, shaded regions, and other shapes within the rectangle.
A
A′
The Complement of a Set
A
A′
The set A′, the shaded region, is the complement of A A′^ is the set of all objects in the universe of discourse that are not elements of A
A′^ = {x | x ∈ U and x ∈/ A}
Set Equality
Is the left set equal to, a proper subset of, or not a subset of the set on the right?
{ 1 , 2 , 3 } I
{a, b} {a} {a} {a, b} {a, b, c} {a, d, e, g} {a, b, c} {a, a, c, b, c} {∅} {a, b, c} {∅} {}
Cardinality of the Power Set
Power Set: P(A) is the set of all possible subsets of the set A
For example, if A = { 0 , 1 }, then P(A) = {∅, { 0 }, { 1 }, { 0 , 1 }}
Find the following power sets and determine their cardinality. P(∅) =
P({a}) =
P({a, b}) =
P({a, b, c}) =
Is there a pattern?
Another Method for Generating Power Sets
A tree diagram can be used to generate P(A). Each element of the set is either in a particular subset, or it’s not.
∅ {c}
{b}
{b} {b, c}
{a}
{a}
{a} {a, c}
{a, b}
{a, b} {a, b, c}
The number of subsets of a set with cardinality n is 2 n The number of proper subsets is 2 n^ − 1 (Why?)
Set Operations
Intersection The intersection of two sets, A ∩ B, is the set of elements common to both: A ∩ B = {x|x ∈ A and x ∈ B}. In other words, for an object to be in A ∩ B it must be a member of both A and B.
A
B
Find the Following Intersections
{a, b, c} ∩ {b, f, g} =
{a, b, c} ∩ {a, b, c} =
For any A, A ∩ A =
{a, b, c} ∩ {a, b, z} =
{a, b, c} ∩ {x, y, z} =
{a, b, c} ∩ ∅ =
For any A, A ∩ ∅ =
For any A, A ∩ U =
For any A ⊆ B, A ∩ B =
Disjoint Sets
Disjoint sets: two sets which have no elements in common.
I.e., their intersection is empty: A ∩ B = ∅
A
B
Are the Following Sets Disjoint?
{ a, b, c } and { d, e, f, g }
{ a, b, c } and { a, b, c }
{ a, b, c } and { a, b, z }
{ a, b, c } and { x, y, z }
{ a, b, c } and ∅
For any A, A and ∅
For any A, A and A′
Set Union
The union of two sets, A ∪ B, is the set of elements belonging to either of the sets: A ∪ B = {x|x ∈ A or x ∈ B}
In other words, for an object to be in A ∪ B it must be a member of either A or B.
A
B
Find the Following Unions
{ a, b, c } ∪ { b, f, g } =
{ a, b, c } ∪ { a, b, c } =
For any A, A ∪ A =
{ a, b, c } ∪ { a, b, z } =
{ a, b, c } ∪ { x, y, z } =
{ a, b, c } ∪ ∅ =
For any A, A ∪ ∅ =
For any A, A ∪ U =
For any A ⊆ B, A ∪ B =
Set Difference
The difference of two sets, A − B, is the set of elements belonging to set A and not to set B: A − B = {x|x ∈ A and x /∈ B}
A
B
Note: x /∈ B means x ∈ B′ Thus, A − B = {x|x ∈ A and x ∈ B′} = A ∩ B′
Set Difference Example
Note, in general, A − B 6 = B − A
Given the sets: U = { 1 , 2 , 3 , 4 , 5 , 6 , 9 } A = { 1 , 2 , 3 , 4 } B = { 2 , 4 , 6 } C = { 1 , 3 , 6 , 9 }
Find each of these sets: A ∪ B =
A ∩ B =
A ∩ U =
A ∪ U =
Suppose U = {q, r, s, t, u, v, w, x, y, z}, A = {r, s, t, i, v}, and B = {t, v, x}
Complete the Venn Diagram to represent U , A, and B
A
B
Shade the Diagram for: A′^ ∩ B′^ ∩ C
C
A B
Shade the Diagram for: (A ∩ B)′
A
B
Shade the Diagram for: A′^ ∪ B′
A
B
De Morgan’s Laws
De Morgan’s Laws: For any sets A and B (A ∩ B)′^ = A′^ ∪ B′ (A ∪ B)′^ = A′^ ∩ B′ Using A, B, C, ∩, ∪, −, and ′, give a symbolic description of the shaded area in each of the following diagrams. Is there more than one way to describe each?
C
A B
C
A B
Cardinal Numbers & Surveys
Suppose, U = The set of all students at EIU A = The set of all male 2120 students B = The set of all female 2120 students A ∪ B = Now suppose, n(A) = 97 n(B) = 101
A
B
n(A ∪ B) =
Counting via Venn Diagrams
Suppose, U = The set of all students at EIU A = The set of all 2120 students that own a car B = The set of all 2120 students that own a truck A ∪ B = A ∩ B =
Now suppose, n(A) = 33 n(B) = 27 n(A ∩ B) = 10 n(A ∪ B) =
A
B
Inclusion/Exclusion Principle
For any two finite sets A and B:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
In other words, the number of elements in the union of two sets is the sum of the number of elements in each of the sets minus the number of elements in their intersection.
How many integers between 1 and 100 are divisible by 2 or 5?
Let, A = {n | 1 ≤ n ≤ 100 and n is divisible by 2 } B = {n | 1 ≤ n ≤ 100 and n is divisible by 5 } n(A) = n(B) = n(A ∩ B) = n(A ∪ B) =
Back to Counting with Venn Diagrams
A
B
C
Find the cardinality of the sets:
A (A ∪ B) ∩ C B A′ A ∩ B ∩ C′^ C − B A ∪ B (A ∪ B) ∩ C′
Venn Diagram for 2 Sets
There are four disjoint regions
I
III
II
IV
I: A ∩ B′
II: A ∩ B
III: A′^ ∩ B
IV: A′^ ∩ B′
Venn Diagram for 3 Sets
There are eight disjoint regions
I II III
IV
V
VI
VII
VIII
I: A ∩ B′^ ∩ C′
II: A ∩ B ∩ C′
III: A′^ ∩ B ∩ C′
IV: A ∩ B′^ ∩ C
V: A ∩ B ∩ C
VI: A′^ ∩ B ∩ C
VII: A′^ ∩ B′^ ∩ C
VIII: A′^ ∩ B′^ ∩ C′
Technique for Counting with Venn Diagrams
Designate the universal set
Describe the sets of interest
Draw a general Venn diagram
Relate known information to the sizes of the disjoint regions of the diagram
Infer the sizes of any remaining regions
Using Venn Diagrams to Display Survey Data
Kim is a fan of the music of Paul Simon and Art Garfunkel. In her collection of 22 CDs, she has the following: 5 on which both Simon and Garfunkel sing 8 total on which Simon sings 7 total on which Garfunkel sings 12 on which neither Simon nor Garfunkel sings
S
G
- How many of her CDs feature only Paul Simon?
- How many of her CDs feature only Art Garfunkel?
- How many feature at least one of these two artists?
M E
G
Student Values
Julie Ward, who sells college textbooks, interviewed freshmen on a community college campus to determine what is important to today’s students. She found that Wealth, Family, and Expertise topped the list. Her findings can be summarized as:
n(W) = 160 n(E ∩ F) = 90 n(F) = 140 n(W ∩ F ∩ E) = 80 n(E) = 130 n(E′) = 95 n(W ∩ F) = 95 n[(W ∪ F ∪ E)′] = 10
How many students were interviewed?
W F
E
How many students were interviewed?
Counting Principles
A B C
We have three choices from A to B and two choices from B to C.
How many ways are there to get from A to C through B?
Multiplication Principle
If k operations (events, actions,...) are performed in succession where:
Operation 1 can be done in n 1 ways Operation 2 can be done in n 2 ways .. . Operation k can be done in nk ways
then the total number of ways the k operations can all be performed is: n 1 ∗ n 2 ∗ n 3 ∗ · · · ∗ nk
In other words, if you have several actions to do and you must do them all you multiply the number of choices to find the total number of choices.
Examples
How many outcomes can there be from three flips of a coin?
Action 1: Flip a coin Action 2: Flip a coin Action 3: Flip a coin Total
How many ways are there to form a three letter sequence from the letters in {A, B, C,... , Z}?
Action 1: Pick a letter Action 2: Pick a letter Action 3: Pick a letter Total
How many ways are there to form a three letter sequence from the letters in {A, B, C,... , Z} without repeating any letter?
Action 1: Pick a letter Action 2: Pick an unused letter Action 3: Pick an unused letter Total
How many ways are there to form a license plate that starts with three uppercase letters and end with 3 digits (0–9)?
Action 1: Pick 3 letters Action 2: Pick 3 digits Total
Counting with Trees
Tree diagrams consist of nodes (the circles) and branches that connect some nodes.
The nodes represent the possible “states” of a situation. Branches are the ways or “choices” we have to move to another state. The “top” node is called the root and it represents the starting state. Leaves, nodes with no other nodes under them, represent an ending state.
Counting with Trees
This leads to the following technique: Use a tree diagram to illustrate a situation Count the number of leaves to find the number of possible outcomes
Examples
How many outcomes can there be from three flips of a coin?
start
H
HH
HHH HHT
HT
HTH HTT
T
TH
THH THT
TT
TTH TTT
8 outcomes
