MM150SURVEY OF MATHEMATICSUnit 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 B = Dad Sister

Brother

D = Dad 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

SECTION 2.2: SUBSETS

SECTION 2.2: SUBSETS

SECTION 2.2: SUBSETS

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

1. Determine the elements in A ∩ B ∩ C.

2. Determine the elements in A ∩ B, B ∩ C, and A ∩ C

(not already listed in #1).

3. Place all remaining elements in A, B, C as needed (not

already listed in #1 or #2).

4. 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}

1. A ∩ B ∩ C

2. A ∩ B, B ∩ C, and A ∩ C

3. A, B, C

4. U

U

SECTION 2.4: VENN DIAGRAMS WITH THREE SETS AND VERIFICATION OF EQUALITY OF SETS

De Morgan’s Laws

1. (A ∩ B)´ = A´ U B´

2. (A U B)´ = A´ ∩ B´

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