Look at the following illustrations.. pharmacist laptop policeman sister teacher father iwatch engineer mother ipod doctor cellphone Describe what you see.

B A C D E Red, Orange, Yellow, Green, Violet, Indigo, Blue Verizon, AT&T, T-Mobile, Sprint Waccamaw Andrews Carvers Bay Georgetown Mouth, esophagus, stomach, liver, pancreas, small intestine, large intestine, appendix, rectum, anus B A C Red, Orange, Yellow, Green, Violet, Indigo, Blue North America, South America, Europe, Asia, Africa, Antarctica, Australia D E

SETS

Examples: The set of the states of USA A SET is a well-defined collection of distinct objects. (Usually denoted by CAPITAL letters.) - A SET is well-defined if its contents can be clearly be determined Examples: The set of the states of USA Examples: The countries that are part Rio Olympics 2016 Examples: The set of the SENIOR students of WHS Examples: The set of all real numbers

A set is a well-defined collection of distinct objects A set is a well-defined collection of distinct objects. (Usually denoted by CAPITAL letters.) Is the group of places a set? How about group of boys? Is the group of three best movies a set?

“WELL - DEFINED SET – means it is possible to determine whether a particular object belongs to a particular set.

Example: Tell whether each set is well defined or not 1. The collection of all the Math subjects offered in WHS Well defined 2. The collection of all basketball players not 3.The collection of all schools in Georgetown Well defined 4. The collection of the best teachers not 5. The collection of your favorite shirt not

- The objects in a set are called ELEMENTS of the set. “b is an element of P” “c is not an element of P”

Examples: Chicago Bulls ____ NBA Teams Bugs Bunny ____ Pokemon 24 ____ odd integer George Bush _____ Presidents of the US

A set that contains no element, is denoted by, Null or Empty Set A set that contains no element, is denoted by, EXAMPLES: Set of WHS students taking 3 Math subjects in a school year Set of grade 9 students in Discrete Math

Natural number solutions Indicate the set of natural numbers that satisfies the equation x+2 = 0 Only the number -2 satisfies this equation. Because -2 is not a natural number, the solution set for this equation is

Two Ways to describe a Set Rule Form/Set Builder Notation Elements are described as {x|x …} x such that… Roster Form Elements are listed and separated by commas. Examples: A = {1, 2, 3, 4, 5} Examples: A = {x| x is a nutaral number less than 6} A = {x|x N and x < 6}

4. D = {red, orange, yellow, green, blue, indigo, violet} Example: Describe the following sets in rule form. 1. A = { 3, 5, 7, 9 } A = {x|x is an odd number between 2 and 10} 2. B = {Monday, Tuesday, Wednesday, Thursday, Friday} B = {x|x is a weekday} 3. C = {square, rhombus, parallelogram, rectangle, trapezoid} C = {x|x is a quadrilateral} 4. D = {red, orange, yellow, green, blue, indigo, violet} D = {x|x is a color of the rainbow}

Example: Describe the following sets in roster form. 1. L = {x|x is an even number between 0 and 10} L = { 2, 4, 6, 8 } 2. E = { x|x is a prime number less than 20} E = { 2, 3, 5, 7, 11,13, 17, 19 } 3. F = { x|x is a vowel in the word mathematics} F = { a, e, i } 4. G = { x|x is an integer} G= { …-3, -2, -1, 0, 1, 2, 3, … }

Example: Describe the following sets in roster form. 1. L = {x|x N and 2 ≤ x ≤ 8} L = { 2, 3, 4, 5, 6, 7 }

1. E = { x|x is a vowel in the alphabet} n(H) = Cannot be determined The CARDINAL NUMBER of a set The cardinal number of Set A, symbolized by n(A), is the number of elements in set A. Examples: State the cardinal number of the given set. 1. E = { x|x is a vowel in the alphabet} n(E) = 5 2. F = {0, 1, 2, 3, 4, 5, 6} n(F) = 7 3. G = { } n(G) = 0 4. H = {2, 4, 6, 8, 10, … } n(H) = Cannot be determined

Cardinal number of the Set Set of the Continents of the World ___________ Set of Rational Numbers ___________ Set of Even Integers between 15 and 37 ___________ 11 7 Cannot be determined

Kinds of Sets Finite vs. Infinite Countable number of elements Examples: A = { x | x is a consonant} B = {1,2,3,4,5} C = { x | x is a Math teacher in WHS} The counting of its elements has no end. Examples: A = { x | x is an even number} B = {5,10,15,20, ...} C = { x | x is a negative number} Finite Sets Infinite Sets

Equal vs. Equivalent n(R) = n(S) J = K Same number of elements Example: If R = {c, u, e, t} and S = {e, l, o, v} then n(R) = n(S) All elements are the same Example: If J = {2, 4, 6, 8, 10} K = { x | x is an even number between 0 and 11} then J = K Equal Sets Equivalent Sets

Any sets that are equal must also be equivalent. Not all sets that are equivalent are equal.

The sets of positive and negative numbers are disjoint. Joint vs. Disjoint There is at least one common element. Example: If and then P and Q are joint sets. There is no common element. Example: The sets of positive and negative numbers are disjoint. Joint Sets Disjoint Sets

then the universal set is (symbolized by U) Is the totality of the elements under consideration. Is a set that contains all the elements for any specific discussion If P= {1, 2, 3, 4, …} , N = {0} and Q = {-1, -2, -3, …} then the universal set is U= {…-3, -2, -1, 0, 1, 2, 3,…} or U= {x|x is an integer}

I. Determine if the given set is well defined. Class Work - August 23, 2016 I. Determine if the given set is well defined. 1. The collection of all easiest courses in WHS 2. The set of the elements in the periodic table 3. The collection of all successful professionals 4. The collection of all rational numbers. 5. The collection of letters in the alphabet II. Describe the following sets in roster and rule form. State the cardinality. 6. The collection of natural numbers 7. The collection of prime numbers between 10 and 50 8. The collection of multiples of 8 less than 40 9. The collection of square numbers greater than 1 10. The collection of months with 31 days III. Tell whether the statement is sometimes true, always true or never true. 11. Equal sets are also equivalent sets. 12. If two sets are disjoint, then they are equivalent. 13. The cardinality of a finite set cannot be determined. 14. If two sets are equal then they are also joint sets. 15. The set of all the students in WHS in 2010 has a finite cardinal number.

REVIEW Sets I. Tell whether the statement is sometimes true, always true or never true. 1. If two sets are equivalent then they are equal. 2. The set {1, 2} is equal to the set { 2,1}. 3. The set of decimals and fractions are joint sets. 4. Rational numbers combined with irrational numbers create the universal set of real numbers. II. Describe the following sets in roster and rule form. State the cardinal number of each set. 5. The set of cube numbers greater than 1 6. The set of nonnegative odd integer less than 20 7. The set of natural numbers

REVIEW III. Tell whether the given set/s is/are finite, infinite, equal, equivalent, joint or disjoint. (each item may have more than one answer) 8. A = { 2, 4, 6, 8, 10} and B = {1, 3, 5, 7,9} 9. A = { 0, 1, 2, 3, 4,…} and B = {1, 2, 3, 4, …} 10. A = { 1, 2, 4} and B = {2, 1, 4}

A B A A SUBSETS If A = { 1, 2, 3} and B = {1, 2, 3, 4, 5} then Ø A Set A is a subset of Set B, denoted as A B, if and only if all the elements of Set A are also elements of Set B Example: If A = { 1, 2, 3} and B = {1, 2, 3, 4, 5} then A B A A Every set is a subset of itself. An empty set is always a subset of every set. Ø A

a. A = {hydrogen, gold, silver} B = {hydrogen, gold, iron, silver} Subsets Determine whether set A is a subset of set B. a. A = {hydrogen, gold, silver} B = {hydrogen, gold, iron, silver} b. A = {5, 10, 15, 20, 25, 30} B = {5, 10, 15, 25} c. A = {chocolate, vanilla, rocky road} B = {rocky road, chocolate, vanilla}

Proper Subsets    

Proper Subsets    

Consider: R = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 5, 6} B = {3, 9} C = {0, 1, 2, 10} Is R  R? Is C  R? Is B  R? Is A  R?

Determine whether the following is TRUE or FALSE.  

Example: List all the subsets of M = {2, 3, 4} * The number of subsets of a given set is 2n where n is the number of elements in a set. Example: List all the subsets of M = {2, 3, 4} 1. A = { 2 } 5. E = { 2, 4 } 6. F = { 3, 4 } 2. B = { 3 } 7. G = { 2, 3, 4 } 3. C = { 4 } 4. D = { 2, 3 } 8. H = { }

PLEASE FOLLWING THE SEATING CHART DOOR BOARD PRICE STAFFORD, JASMINE CHARLTON PROJECTOR BARKER BROWN TABLE KEATING GREENE STAFFORD, JOSHUA RATTO PRICE, HUNTER SANGREGORY ROY KOONTS CROOKS VELTRI ROURK TAYLOR LOTTCHEA SMALL PLEASE FOLLWING THE SEATING CHART

Operations on Sets Venn Diagram (by John Venn) is used to represent relationships between sets. U A B 1 2 3 4 5 6 7 8 9 10 1. Union of Sets A and B = set of all elements found in A or B, or both A and B. Example: A U B = { 1, 2, 3, 4, 5, 6, 7}

2. Intersection of Sets A and B U A B 1 2 3 4 5 6 7 8 9 10 = set of all common elements in A and B. Example: A ∩ B = { 4, 5 } 3. Complement of Set A = set of all elements in the Universal set but not found in A

3. Difference of Sets A and B = set of all elements found in A but not in B. U A B 1 2 3 4 5 6 7 8 9 10 = { 1, 2, 3 } = { 6, 7 } Example: Let A = {1, 2, 3, 4} and let B = {3, 4, 5, 6} and C = { 6, 7,8} Find: = { 1, 2} = {1, 2, 3, 4} = { 5, 6} = { 3, 4, 5}

X Y Z 1 5 3 9 8 4 2 U 6 7 Use the Venn Diagram to list the members of the specified sets.

BASIC VENN DIAGRAM SHADING VIDEO PRESENTATION BASIC VENN DIAGRAM SHADING

Shade the region representing each set. A U B U C A U B ∩ C A’ -(B ∩ C) (B U C)- A’

1. What is the universal set? U = {2, 3, 4, 5, 6, p, n, o, y} 2. E’ Consider the ff. sets. Give the elements of the sets being described below. E = {3, 4, 5, 6} , I = {2, 6, p} , J = {y, n, p, y} 1. What is the universal set? U = {2, 3, 4, 5, 6, p, n, o, y} 2. E’ 3. E∩I ∩J 4. (I U J)’ ∩ E 5. (J - I) U E 6. (E U I) – (J ∩ I) = {2, o, n, p, y} = { } = {3, 4, 5} = {3, 4, 5, 6, n, o, y} = {2, 3, 4, 5, 6}

A B C II. List all the subsets of A = {1, 5, 9}. CLASS WORK I. Use the Venn Diagram to list the members of the specified sets. 1. 2. A B C 1 8 6 7 5 2 9 4 3 3. 4. 5. 6. 7. U 11 II. List all the subsets of A = {1, 5, 9}.

III. Shade the region representing each set. B C A B C U U A B C A B C U U

Assignment: ½ crosswise In a survey of 75 consumers, 12 indicated that they were going to buy a new car, 18 said they were going to buy a new refrigerator, and 24 said they were going to buy a new washer. Of these, 6 were going to buy both a car and a refrigerator, 4 were going to buy a car and a washer, and 10 were going to buy a washer and a refrigerator. One person indicated that she was going to buy all three items. Construct a Venn diagram, label your diagram clearly.  Use your diagram to answer the following questions: (a) How many were going to buy only a car? (b) How many were going to buy only a washer? (c) How many were going to buy only a refrigerator (d) How many were going to buy a car and a washer but not a refrigerator? (e) How many were going to buy none of these items?