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Chapter Two

• Basic Concepts of Set Theory– Symbols and Terminology– Venn Diagrams and Subsets

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What is a Set?

• Set is a collection of Objects

Objects belonging to the set are called elements of the set, or members of the set.

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Sets are described in three ways

– Word descriptions

The set of even counting numbers less than ten

– Listing method

{2,4,6,8}

– Set-builder notation

{ X| X is an even counting number less than 10}

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Suppose E is the name for the set of all letters of the alphabet. Then we can write

E = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}

• We can shorten a listing by using ellipsis points. For example:E = {a,b,c,d,…,x,y,z}

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Examples

List the elements of the set of each of the following:

A) The set of counting number between six and thirteen

Answer: { 7, 8, 9, 10,11,12}

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B) List each element of the set{ 5,6,7,…10}

Answer: Completing the list we get {5,6,7,8,9,10}

C) {X | X is a counting number between 6 and 7}

Answer: There are no elements – so we write

{ } or 0

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Empty or Null Set

• Empty set is denoted 0 or { }

• Do not use { 0 } or { 0 } to denote the empty set.

• Empty set is denoted 0 or { }

• Do not use { 0 } or { 0 } to denote the empty set.

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Sets of Numbers

• Natural or Counting Numbers{1,2,3,4,…}

• Whole Numbers{0,1,2,3,4,…}

• Integers{…,-1,0,1,….}

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• Rational Numbers{p/q | p and q are integers and q not equal

to 0.(ex. ¾, -7/5, ½ or .55, .67 etc….)

• Real Numbers{x | x can be written as a decimal }

• Irrational Numbers{x | x is a real number and x cannot be written as a quotient of integers}

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Cardinal Numbers

• The number of elements in a set is called the cardinal number, or cardinality of the set. The symbol n(A) is read “n of A” and represents the cardinality of set A.

• If elements are repeated in a set, they should not be counted more than once when determining the cardinality of the set.

For example, if the set, B = { 1,1,2,2,3,3} there are three distinct elements in the set and

n(B) = 3

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Examples

• Find the cardinal number of each of the following sets:

1. K = {2,4,8,16} n(K) =

2. M = {0} n(M) =

3. R = { 4,5,…,12,13} n(R) =

4. Empty set 0 n(0) =

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1. K = {2,4,8,16} n(K) = 4

2. M = {0} n(M) = 1

3. R = { 4,5,…,12,13} n(R) = 10

4. Empty set 0 n(0) = 0

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Finite and Infinite Sets

• If the cardinal number of a set is a whole number or a counting number – then that set is finite set. We can count it.

• Example: B = { 1,2,3,4,5,6,7,8,9,10}

• Some sets are so large we cannot count the elements in the set.

• If the set is so large that its cardinal number is not found among the whole numbers, we call that an infinite set.

• For example the set of counting numbers is an infinite set.

• Example B = {1,2,3,4,….}

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Exercise

• Review – what are the three common ways to write set notation?

• Word Description• Listing Method• Set Builder Notation

• Now, write the set of all odd counting numbers using a word description, listing method, and set builder notation

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Set Equality

• Set A is equal to set B provided the following two conditions are met:

1. Every element of A is an element of B

and

2. Every element of B is an element of A.

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Examples• True or False ….

• {a,b,c,d} = {d,c,b,a}

• {1,0,1,2,3,} = {0,1,2,3}

• {4,3,2,-1} = {3,2,4,1}

• {4,3,2,-1} = {3,2,4,1}

True

True

True

False

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Venn Diagrams and Subsets

• Universe of Discourse– For a problem includes all things under

discussion at a given time.

Suppose the NOVA Loudon campus considered raising the scores for the Algebra 1 placement exam. The universe of discourse might be all potential students wishing to take Algebra 1 from the Loudon campus.

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Universal Set

• In mathematical theory of sets, the universe of discourse is known as the Universal Set.

• The letter U is usually used for the universal set.

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Venn Diagrams

• The universal set is represented by a rectangle, and other sets of interest within the universal set are represented by an oval region, circles, or other shapes.

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Venn Diagrams

U

A

A’The entire region bounded by the rectangle represents the Universal Set - U

The oval represents the Set A

The region inside U and outside the oval is labeled A’ (read A prime)

This is the compliment of A

Contains elements in U not in A

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Compliment of a Set

• For any set A within the universal set U, the complement of A, written A’ is the set of elements of U that are not elements of A .

A’ = { X | X E U and X E A}

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Subset of a Set

• How do we define the compliment of the universal set, U’.

• The set U’ is found by selecting all the elements of U that do not belong to U.

U

A

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• For the universal set U• U’ = 0

• Next, lets look at the compliment of the empty

set, 0’.• Since 0’ = { X | X E U and X E 0 } and set 0

contains no elements, every member of the universal set U satisfies this definition

U

A

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• So, for every universal set U, 0’ = U

• Suppose U = {1,2,3,4,5}

• Let A = {1,2,3}

• Every element of A is an element of set U

• Set A is called a subset of set U

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Subset of a Set

• Set A is a subset of set B if every element of A is also an element of B.

• A B

Examples

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Set Equality

• If A and B are sets, then A = B if

• A B and B A.

Suppose B = { 5,6,7,8} and A= {6,7}.

The A is a subset of B, but A is not all of B.

A is called a proper subset of B.

A B.

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Proper Subset of a Set

• Set A is a proper subset of set B if

A B and A = B

Then A B.

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• Set A is a subset of set B, if every element of set A is also an element of set B.

• Set A is a subset of Set B, if there are no elements of A that are not also element of B

• IS the empty set a subset of A or B or both?• 0 B

– The empty set is a proper subset of every set except itself

– Every set (except the empty set) has at least two subsets, the set itself and the empty set.

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Finding the number of subsets

• Number of Subsets– The number of subset of a set with n elements is 2 n

– Since the value 2 n contains the set itself, we must subtract 1 from this value to obtain the number of proper subsets of a set containing n elements.

• The number of proper Subset of a set with n elements is 2n -1

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Homework• Exercises • 2.1 Page 54

9 -21 odd, 25, 27, 29, 31, 33, 35, 41 -49 odd, 59 – 66 odd,67 – 78 odd

2.2 Page 617 -41 odd, 43, 45, 49, 52