Unit 2: Sets

Prof. Carolyn Dupee

July 3, 2012

HOW DO YOU WRITE SETS?

•P. 69 Ex. 2•Set A is the set of all natural numbers (counting numbers) less than 5.

- A= {1,2,3,4}

•Set B is the set of natural numbers less than or equal to 75.- B= {1,2,3,4, . . . . . .75} . . . Means the numbers keep going

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EXPRESS THE FOLLOWING IN ROSTER FORM.

•The set of natural numbers between 4 and 9.-A= {4,5,6,7,8,9}

•The set of natural numbers between 4 and 9, inclusive.-B= {4,5,6,7,8,9}

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SET BUILDER NOTATION

D = {x | x N and x> 10}

Set D is x such that x is a natural number and x is greater than 10.

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WRITE SET THE SET:

•Set c= {North America, South America, Asia, Australia, Africa, Antartica} in set builder notation

-A= {x | x is a continent}•Write in words how you would read set C in set builder notation.

-Set C is the set of all elements x such that x is a continent.

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COMPARING SETS

•Set A = {1,2,3} Set B = {3,2,1}

-Set A equals set B because the two sets have the same elements.

-Set A is equivalent to set B if the have the same cardinality (# of things in the set).

- { } would be an empty set with no numbers in it

•Poll Question 1: - Set S= {North Carolina, Georgia, South Carolina, Florida}- -Set C= {Columbia, Raleigh, Tallahasse, Atlanta}

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SECTION 2.2 SUBSETS

•Determine whether set A is a subset of set B.•Problem 1: A= {marigold, pansy, geranium}•B= {marigold, pansy, begonia, geranium}

- All of the elements of set A are contained in set B, so A ⊆

•Problem 2: A= {2,3,4,5} B= {2,3}-The elements 4 and 5 are in set A, but not in set B, so A ⊆ B (A is not a subset of B). In this example, however, all the elements of set B are contained in set A; therefore, B ⊆ A.

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SECTION 2.2: SUBSETS CONT.

•Problem 3: A: {x| x is a yellow fruit} B= {x | x is a red fruit}- There are fruits, such as bananas, that are in set A that are not in set B, so A ⊆ B.

• (Poll question 2)•Problem 4: A= {vanilla, chocolate, rocky road} •B= {chocolate, vanilla, rocky road}

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PROPER SUBSET

•A ⊂ B, iff (if and only if) all the elements of set A are elements of set B and set A doesn’t equal set B (Set B must contain at least one element not in set A).

•Ex. 2, p. 78- Determine whether set A is a proper subset of set B.- A= {jazz, pop, hip hop}- B= {classical, jazz, pop, rap, hip hop}

-All elements of set A are contained in set B, and sets A and B are not equal so A ⊂ B.

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PROPER SUBSETS- POLL 3

•A= {a, b, c, d} B= {a, c, b, d}• Is set A a proper subset of set B?

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NUMBER OF SUBSETS (CARDINALITY) P. 79

•Formula for number of subsets= 2n, where n is the number of elements in the set A.

- { } 20= 1- { a } 21= 2- {a, b} 22= 4- {a, b, c}= 23= 8

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NUMBER OF SUBSETS (CARDINALITY EX. 4)

• Determine the number of distinct subsets for the set {S,L,E,D}

- 2n= 24=16

• List all the distinct subsets for the set {S,L,E,D}

- {S,L,E,D} {S,L,E}; {S,L,D}; {S,E,D}; {L,E,D}

- {S,L}; {S,E}; {S,D}; {L,E}, {L, D}; {E,D} {S}; {L}, {E}, {D} { }

• How many of the distinct subsets are proper subsets?

- There are 15 proper subsets. {S,L,E,D} can’t be a proper subset of itself.

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UNDERSTANDING VENN DIAGRAMS

•Example 1, p. 84: Given U (universal set) = {1,2,3,4,5,6,7,8} and A= {1,3,4} find A’ (complement or NOT A) and illustrate the relationship among sets U,A, and A’ in a Venn Diagram.

1. What’s in set A? 1, 3, 4

2. What’s not in set A or in A’? 2, 5, 6, 7, 8

3. What doe the sets U and A have in common? 1,3,4

4. Draw the Venn Diagram (See next slide)

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UNDERSTANDING VENN DIAGRAMS CONT.

14

U

1 3

4

A

A’

2,5,6,7,8

INTERSECTION OF VENN DIAGRAMS

•A ∩ B where elements are shared by sets A and B.•Example 2, p. 85: U= 50 states in the United States, B= set of states with a population of more than 10 million people as of 2005, B= set of states that have at least one city with a population more than 1 million people as of 2005. Draw a Venn Diagram illustrating the relationship between sets A and set B.

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INTERSECTION OF VENN DIAGRAMS ∩ STEP 1

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Population > 1 million One city > 1 million

California CA California CA

Texas TX Texas TX

New York NY New York NY

Florida FL

Illinois IL Illinois IL

Pennsylvania PA Pennsylvania PA

Ohio OH Arizona AZ

Michigan MI

INTERSECTION OF VENN DIAGRAMS ∩ STEP 2

U

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A B

I IICATXNYIL PA

III

AZOHMI FL All other

U.S. states

INTERSECTION OF SETS NOTATION PRACTICE

•U = {1,2,3,4,5,6,7,8,9,10} A={1,2,3,8} B= {1,3,6,7,9} C= { }•Question A) A ∩ B= What’s in common to both A and B? A AND B

- Answer: A={1,2,3,8} B= {1,3,6,7,9} The elements 1 and 3.- Notation {1,3)

•Question B) A ∩ C= What’s in common to both A and C? A AND C- Answer: A={1,2,3,8} C= { } - This is a trick question, the elements are inside the brackets and they don’t have

any elements in common in this case.

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CONT. INTERSECTION OF SETS PRACTICE

•U = {1,2,3,4,5,6,7,8,9,10} A={1,2,3,8} B= {1,3,6,7,9} C= { }•Question C) A’ ∩ b= What’s common to NOT A and B? Not A and B

- Answer: not A= 4,5,6, 7,9 B= 1,3,6,7,9

- Notation: {6,7,9}

•Question D) (A ∩ B)’ - A ∩ B = 1, 3- NOT A ∩ B = NOT A AND NOT in B - {2,4,5,6,7,8,9,10}

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UNION (OR)

•Where do the elements unite?•A B ∪•P. 86 Ex. 4: Use the Venn Diagram to determine the sets. (See next slide).

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UNION PRACTICE (P. 86 EX. 4)

U A B

I IIIII

9 3 ?

7

IV

8 #

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UNION PRACTICE P. 86, EX. 4

•A) What’s the Universal Set?- U= {9, triangle, 8, #, square, circle, 3, ?, 7}- You can use the symbols when you’re doing MML graded practice they will show

up in your box to the left.

•B) What’s in set A?- A= {9, triangle, square, circle}

•C) What’s the complement of set B (B’)?- {9, triangle, 8, #}

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UNION PRACTICE P. 86, EX. 4

•D) A ∩ B- {square, circle}

•E) A ∪ B- {9, triangle, square, circle, 3, ?, 7}

•F) (A B)’ = {#, 8}∪•G) n (A B) ∪ This is asking for the number of elements.

- 2 + 2+ 3 = 7

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USING VENN DIAGRAMS WITH 3 CIRCLES

•P. 96, Ex. 1: Construct a Venn Diagram illustrating the following sets:

- U= {1,2,3,4,5,6,7,8,9,10,11,12,13,14}- A= {1,5,8,9,10,11}- B= {2,4,5,9,10,13}- C= {1,3,5,8,9,11}

•What’s in common?- A + C: 1, 5, 8 -A + B + C: 5, 9 -A + B: 5, 9, 10 B + C: 5, 9- Not in any only U: 6, 7, 14

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DRAWING VENN DIAGRAMS WITH 3 CIRCLES

U A B Be ready for some poll questions!!

I II III 13

12 10 2 4

IV 5,9

1 8 V VI

VII

3 11 VIII 6,7,14

C

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