1 Chapter Two Basic Concepts of Set Theory –Symbols and Terminology –Venn Diagrams and Subsets.
Transcript of 1 Chapter Two Basic Concepts of Set Theory –Symbols and Terminology –Venn Diagrams and Subsets.
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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