Section 2.3 Set Operations and Cartesian Products Intersections of Sets The intersection of Set A and B, written is the set of elements common to both A and B.

Example Suppose we have two candidates, Mr. Brown and Mr. Green running for a city office. A voter decided for whom she should vote by recalling their campaign promises. Mr. Green Mr. Brown Spend less money, m Emphasize traffic law enforcement, t Increase services to suburban areas, s Crack down on crooked politicians, p Increase services to city, c

Mr. Green Mr. Brown Spend less money, m Emphasize traffic law enforcement, t Increase services to suburban areas, s Crack down on crooked politicians, p Increase services to city, c Lets look at each candidates promises as a set. Mr. Green’s set = { m,t,s} Mr. Brown’s set = { m,p,c} The only element common to both sets is m, this is the intersection of both sets.

To represent the sets as a Venn Diagram Mr. Brown Mr. Green T S P C M

Find the intersection of the given sets A) { 3,4,5,6,7} and {4,6,8,10} Elements common to both sets: { 3,4,5,6,7} {4,6,8,10} = {4,6} B) { 9,14,25,30} {10,17,19,38,52} { 9,14,25,30} {10,17,19,38,52} =

C) { 5,9,11} and { 5,9,11} = Sets with no elements in common are called disjoint sets A set of dogs and a set of cats are disjoint sets DOGS CATS

Union of Sets Form the union of sets A and B by taking all the elements of set A and including all the elements of set B. Set B Set A P C T S M

Union of Sets

Find the union of the sets A) { 2,4,6} and {4,6,8,10,12} Answer: { 2,4,6} {4,6,8,10,12} = {2,4,6,8,10,12} B) { a,b,c,d} and { c, f, g} Answer: {a,b,c,d,f,g} C) {3,4,5} and Answer: {3,4,5}

More examples Find U = { 1,2,3,4,5,6,9} A = { 1,2,3,4} B = { 2,4,6} C = { 1,3,6,9} Find

Find

Answer:

Try to describe the following sets in words: 1. The set of all elements that are in A, and are in B or not in C. The set of all elements that are not in A or not in C, and are not in B.

Differences of Sets Let Set A = { 1,2,3,4,5,6,7,8,9,10} Let Set B = { 2,4,6,8,10} If the elements from B are taken away from Set A then Set C = {1,3,5,7,9} Set C is the difference of sets A and B.

Difference of Sets The difference of sets A and B, written A – B is the set of all elements belonging to set A and not to set B, or

A - B A B

Examples Let U = {1,2,3,4,5,6,7} Find A = {1,2,3,4,5,6} B = { 2,3,6} C = {3,5,7} Find A – B B – A (A – B)

Begin with set A and exclude any elements found also in set B. Let U = {1,2,3,4,5,6,7} A = {1,2,3,4,5,6} B = { 2,3,6} C = {3,5,7} Find A – B Begin with set A and exclude any elements found also in set B. So A – B = {1,2,3,4,5,6} – { 2,3,6 } = {1,4,5}

For B-A an element must be in set B and not in set A. Let U = {1,2,3,4,5,6,7} A = {1,2,3,4,5,6} B = { 2,3,6} C = {3,5,7} Find B – A For B-A an element must be in set B and not in set A. But all elements of B are in A, so B-A = {2,3,6} – {1,2,3,4,5,6} = B 2,3,6 A 1,4,5

In general A – B does not equal B - A Let U = {1,2,3,4,5,6,7} A = {1,2,3,4,5,6} B = { 2,3,6} C = {3,5,7} Find (A – B) We know A – B = {1,4,5} And C’ = { 1,2,4,6} So (A – B) = { 1,2,4,5,6} In general A – B does not equal B - A

Writing ordered pairs In set notation {4,5} = {5,4} There are many instances in math where order matters. So we write ordered pairs using parentheses. Ordered Pairs: In the ordered pair (a,b), a is called the first component and b is called the second component. In general (a,b) (b,a).

Ordered Pairs Two ordered pairs (a,b) and (c,d) are equal if their first components are equal and if their second components are equal. So (a,b) = (c,d) if and only if a = c and b=d. True or false: (4,7) = (7,4)

Cartesian Product of Sets A set may contain ordered pairs as elements. If A and B are sets, then each element of A can be paired with an element of B. The set of all ordered pairs is known as the Cartesian Product of A and B. Written A X B.

Exercises Find A X B = B X A = A X A = For Set A = {1,5,9} and B = { 6,7} {(1,6),(1,7),(5,6),(5,7),(9,6),(9,7)} {(6,1),(6,5),(6,9),(7,1),(7,5),(7,9)} {(1,1),(1,5),(1,9),(5,1),(5,5),(5,9),(9,1),(9,5),(9,9)}

Cardinal Number of a Cartesian Product For example: Set A = {1,2,3}, then n(A) = Set B = {4,5}, then n(B)= 3 2 So, n(A) X n(B) = 3x2 = 6 AXB= {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}

Set Operations Let A and B be any sets, with U the universal set. The complement of A, written A’ is A’ A U

The intersection of A and B is U

The Union of A and B is A B U

The Difference of A – B is U

The Cartesian Product of A and B is

Regions of Venn Diagrams Region 1: Elements outside of set A and Set B. Region 2: Elements belong to A but not to B Region 3: Elements belonging to both A and B Region 4: Elements belong to B but not to A U A B 1 3 4 2 1 A B 3 2 4 U

Example 1 B 3 2 4 A Let U = {q,r,s,t,u,v,w,x,y,z} Let A = {r,s,t,u,v} Let B = {t,v,x} Place the elements in their proper regions. U A B 1 3 4 2 q, w, y, z x t, v First find intersection points r, s, u What elements belong to B and A? What elements belong to U?

Venn Diagram Represent three sets. Let A, B and C be sets 1 B A 6 4 2 5 7 3 C 8 U

Exercises Shade the set Is the statement true

Use the Venn Diagram 1 Region 3 B A Region 1,2,4 3 3 2 4 Region 1,4 U

De Morgan’s Laws For any sets A and B,

Section 2.4 Cardinal Numbers and Surveys Suppose we have this data from a survey 33 people like Tim McGraw 32 favor Celine Dion 28 favor Britney Spears 11 favor Tim and Celine 15 favor Tim and Britney 5 like all performers 7 like non of the performers Can we determine the total number of people surveyed from the data.

First look at intersection region d – 5 who like all three singers 7 7 like non Region a Region b 33-10-5-6=12 a b c d e f g h Britney Spears Tim Mc Graw 10 11 like Tim and Celine Put them in regions d and e 11-5=6 for region e Region g 14 like Britney and Celine 14-5=9 12 4 5 9 6 Region h 28-10-5-9= 4 15 like Britney and Tim Regions c and d 15 -5 = 10 region c 12 Region f 32-6-5-9=12 Celine Dion U To find out how many students were surveyed – Add numbers in all the regions - 65

Cardinal Number Formula For any two sets A and B

Find n(A) Try this Use the formula: Rearrange the formula to find n(A)

Some utility company has 100 employees with T = set of employees who can cut trees P = set of employees who can climb poles W = set of employees who can splice wires 3 17 14 13 23 T First do intersection of T,P,W = 11 Next find P intersection with W 20-11=9 Next find T intersection with W 25-11= 14 Next T intersection with P 28-11=17 T is 45 – 14-11-17 =3 W 3 P 17 11 13 14 9 W 23 U

Homework Section 2.3 odd only 7-28,41-54,55-96,97,101,105,109,117,127 1-16,17