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MM150 Survey of Mathematics. Unit 2 Seminar - Sets. 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. Section 2.1: Set Concepts. Example : The set of months in the year is:

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mm150 survey of mathematics

MM150Survey of Mathematics

Unit 2 Seminar - Sets

section 2 1 set concepts
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.
section 2 1 set concepts1
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:

section 2 1 set concepts2
Section 2.1: Set Concepts
  • The symbol Є means “is an element of”.

Example: March Є { January, February, March, April }

Example: Kaplan Є { January, February, March, April }

section 2 1 set concepts3
Section 2.1: Set Concepts
  • Set-builder notation doesn’t 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 Є N and x < 7 }

section 2 1 set concepts4
Section 2.1: Set Concepts

Sample: A = { x | x Є N and x < 7 }

Example: Write the following using Set Builder Notation.

K = { 2, 4, 6, 8 }

section 2 1 set concepts5
Section 2.1: Set Concepts

Sample : A = { x | x Є N and x < 7 }

Example: Write the following using Set Builder Notation.

S = { 3, 5, 7, 11, 13 }

section 2 1 set concepts6
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?

section 2 1 set concepts7
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)?

section 2 1 set concepts8
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?

section 2 1 set concepts9
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}!

section 2 1 set concepts10
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.

section 2 2 subsets
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

section 2 2 subsets1
Section 2.2: Subsets

Mom

B = Dad Sister

Brother

D =Dad Brother

section 2 2 subsets2
Section 2.2: Subsets

7

3

B = 4 5

1

13

3 1

A =1 C = 6

4 13

section 2 2 subsets3
Section 2.2: Subsets

12

4

B = 8 6

2

10

4 10

A =2 6 C = 6

12 8 8

10

section 2 2 subsets4
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?

section 2 2 subsets5
Section 2.2: Subsets
  • Set A is a propersubset of set B, symbolized by A B, if and only if all the elements of set A are in set B andset A ≠ set B.

A =1, 2, 3

B =1, 2, 3, 4, 5

C =1, 2, 3

section 2 2 subsets6
Section 2.2: Subsets
  • Set A is a propersubset 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?

section 2 2 subsets7
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?

section 2 2 subsets8
Section 2.2: Subsets

Example: List the subsets of A.

A = { a, b, c }

section 2 3 venn diagrams and set operations
Section 2.3: Venn Diagrams and Set Operations
  • A Venn diagram is a picture of our sets and their relationships.
section 2 3 venn diagrams and set operations1
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´ ?

section 2 3 venn diagrams and set operations2
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´ ?

section 2 3 venn diagrams and set operations3
Section 2.3: Venn Diagrams and Set Operations
  • The intersection of sets A and B, symbolized by A ∩ 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 ∩ B?

B ∩ C?

C ∩ A?

section 2 3 venn diagrams and set operations4
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?

section 2 3 venn diagrams and set operations5
Section 2.3: Venn Diagrams and Set Operations
  • Special Relationship:

n(A U B) = n(A) + n(B) - n(A ∩ B)

B = { Max, Buddy, Jake, Rocky, Bailey }

G = { Molly, Maggie, Daisy, Lucy, Bailey }

section 2 3 venn diagrams and set operations6
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 Є N, n is odd }

B = { n | n Є N, n > 10 }

What is A - B?

section 2 4 venn diagrams with three sets and verification of equality of sets
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 ∩ B ∩ C.
  • Determine the elements in A ∩ B, B ∩ C, and A ∩ 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.
section 2 4 venn diagrams with three sets and verification of equality of sets1
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 ∩ B ∩ C
  • A ∩ B, B ∩ C, and A ∩ C
  • A, B, C
  • U

U

section 2 4 venn diagrams with three sets and verification of equality of sets2
Section 2.4: Venn Diagrams with Three Sets and Verification of Equality of Sets

De Morgan’s Laws

  • (A ∩ B)´ = A´ U B´
  • (A U B)´ = A´ ∩ B´
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