MM150 Survey of Mathematics

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MM150Survey of Mathematics

• Unit 2 Seminar - Sets

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Section 2.1 Set Concepts

• A set is a collection of objects.
• The objects in a set are called elements.
• Roster form lists the elements in brackets.

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Section 2.1 Set Concepts

• Example The set of months in the year is
• M January, February, March, April, May, June,
  July, August, September, October, November,
  December
• Example The set of natural numbers less than
  ten is

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Section 2.1 Set Concepts

• The symbol ? means is an element of.
• Example March ? January, February, March,
  April
• Example Kaplan ? January, February, March,
  April

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Section 2.1 Set Concepts

• Set-builder notation doesnt list the elements.
  It tells us the rules (the conditions) for being
  in the set.
• Example M x x is a month of the year
• Example A x x ? N and x lt 7

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Section 2.1 Set Concepts

• Sample A x x ? N and x lt 7
• Example Write the following using Set Builder
  Notation.
• K 2, 4, 6, 8

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Section 2.1 Set Concepts

• Sample A x x ? N and x lt 7
• Example Write the following using Set Builder
  Notation.
• S 3, 5, 7, 11, 13

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Section 2.1 Set Concepts

• Set A is equal to set B if and only if set A and
  set B contain exactly the same elements.
• Example A Texas, Tennessee
• B Tennessee, Texas
• C South Carolina, South Dakota
• What sets are equal?

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Section 2.1 Set Concepts

• The cardinal number of a set tells us how many
  elements are in the set. This is denoted by
  n(A).
• Example A Ohio, Oklahoma, Oregon
• B Hawaii
• C 1, 2, 3, 4, 5, 6, 7, 8
• What is n(A)?
• n(B)?
• n(C)?

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Section 2.1 Set Concepts

• Set A is equivalent to set B if and only if n(A)
  n(B).
• Example A 1, 2
• B Tennessee, Texas
• C South Carolina, South Dakota
• D Utah
• What sets are equivalent?

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Section 2.1 Set Concepts

• The set that contains no elements is called the
  empty set or null set and is symbolized by or
  Ø.
• This is different from 0 and Ø!

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Section 2.1 Set Concepts

• The universal set, U, contains all the elements
  for a particular discussion.
• We define U at the beginning of a discussion.
• Those are the only elements that may be used.

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Section 2.2 Subsets

• Set A is a subset of set B, symbolized by A
  B, if and only if all the elements of set A are
  also in set B.
• orange
• yellow
• B red purple
• blue
• green

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Section 2.2 Subsets

• Mom
• B Dad Sister
• Brother
• D Dad Brother

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Section 2.2 Subsets

• 7
• 3
• B 4 5
• 1
• 13
• 3 1
• A 1 C 6
• 4 13

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Section 2.2 Subsets

• 12
• 4
• B 8 6
• 2
• 10
• 4 10
• A 2 6 C 6
• 12 8 8
• 10

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Section 2.2 Subsets

• Set A is a subset of set B, symbolized by A
  B, if and only if all the elements of set A are
  also in set B.
• Example A Vermont, Virginia
• B Rhode Island, Vermont, Virginia
• Is A B?
• Is B A?

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Section 2.2 Subsets

• Set A is a proper subset of set B, symbolized by
  A B, if and only if all the elements of set
  A are in set B and set A ? set B.
• A 1, 2, 3
• B 1, 2, 3, 4, 5
• C 1, 2, 3

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Section 2.2 Subsets

• Set A is a proper subset of set B, symbolized by
  A B, if and only if all the elements of set
  A are in set B and set A ? set B.
• Example A a, b, c
• B a, b, c, d, e, f
• C a, b, c, d, e, f
• Is A B?
• Is B C?

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Section 2.2 Subsets

• The number of subsets of a particular set is
  determined by 2n, where n is the number of
  elements.
• Example A a, b, c
• B a, b, c, d, e, f
• C
• How many subsets does A have?
• B?
• C?

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Section 2.2 Subsets

• Example List the subsets of A.
• A a, b, c

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Section 2.3 Venn Diagrams and Set Operations

• A Venn diagram is a picture of our sets and their
  relationships.

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Section 2.3 Venn Diagrams and Set Operations

• The complement of set A, symbolized by A', is the
  set of all the elements in the universal set that
  are not in set A.
• Example U m m is a month of the year
• A Jan, Feb, Mar, Apr, May, July, Aug, Oct,
  Nov
• What is A ?

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Section 2.3 Venn Diagrams and Set Operations

• The complement of set A, symbolized by A', is the
  set of all the elements in the universal set that
  are not in set A.
• Example U 2, 4, 6, 8, 10, 12
• A 2, 4, 6
• What is A ?

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Section 2.3 Venn Diagrams and Set Operations

• The intersection of sets A and B, symbolized by A
  n B, is the set of elements containing all the
  elements that are common to both set A and B.
• Example A pepperoni, mushrooms, cheese
• B pepperoni, beef, bacon, ham
• C pepperoni, pineapple, ham, cheese
• What is A n B?
• B n C?
• C n A?

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Section 2.3 Venn Diagrams and Set Operations

• The union of sets A and B, symbolized by A U B,
  is the set of elements that are members of set A
  or set B or both.
• Example A Jan, Mar, May, July, Aug, Oct,
  Dec
• B Apr, Jun, Sept, Nov
• C Feb
• D Jan, Aug, Dec
• What is A U B?
• B U C?
• C U D?

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Section 2.3 Venn Diagrams and Set Operations

• Special Relationship
• n(A U B) n(A) n(B) - n(A n B)
• B Max, Buddy, Jake, Rocky, Bailey
• G Molly, Maggie, Daisy, Lucy, Bailey

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Section 2.3 Venn Diagrams and Set Operations

• The difference of two sets A and B, symbolized by
  A B, is the set of elements that belong to set
  A but not to set B.
• Example A n n ? N, n is odd
• B n n ? N, n gt 10
• What is A - B?

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Section 2.4 Venn Diagrams with Three Sets and
Verification of Equality of Sets

• Procedure for Constructing a Venn Diagram with
  Three Sets A, B, and C
• Determine the elements in A n B n C.
• Determine the elements in A n B, B n C, and A n
  C (not already listed in 1).
• Place all remaining elements in A, B, C as needed
  (not already listed in 1 or 2).
• Place U elements not listed.

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Section 2.4 Venn Diagrams with Three Sets and
Verification of Equality of Sets

• Venn Diagram with Three Sets A, B, and C
• U 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
• A 2, 4, 6, 8, 10
• B 1, 2, 3, 4, 5
• C 2, 3, 5, 7, 8
• A n B n C
• A n B, B n C, and A n C
• A, B, C
• U

U
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Section 2.4 Venn Diagrams with Three Sets and
Verification of Equality of Sets

• De Morgans Laws
• (A n B) A U B
• (A U B) A n B

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Thank You!

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• Use the MML Graded Practice
• Read the DB
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