MM150 Survey of Mathematics

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# MM150 Survey of Mathematics - PowerPoint PPT Presentation

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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### MM150Survey 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:

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 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 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 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 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 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 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 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 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 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
• 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: Subsets

Mom

Brother

Section 2.2: Subsets

7

3

B = 4 5

1

13

3 1

A =1 C = 6

4 13

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: 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: 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: 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: 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: Subsets

Example: List the subsets of A.

A = { a, b, c }

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 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 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 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 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 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 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

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 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 Sets

De Morgan’s Laws

• (A ∩ B)´ = A´ U B´
• (A U B)´ = A´ ∩ B´
Thank You!