Chapter 7: Sets and Venn diagrams Goal: TLW apply set notation. Opening Problem A city has three football teams in the national league: A,

B and C In the last season, 20% of the citys population saw team A play, 24% saw team B and 28% say team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play. How could we represent this information more simply on a diagram? What percentage of the population:

Saw only team A play? Saw team A or team B play but not team C? Did not see any of the teams play? Set: a collection of numbers or objects V={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Element: a member of the set Notation:

1 V 10 V Empty Set: contains no element or {} Special Number Sets Natural/Counting Numbers (): {0, 1, 2, 3, } Integers (): {0,

Positive Integers Negative Integers Rational (): all numbers that can be written as a fraction , 5, Real () all numbers on the number line.

Counting Elements of Sets n(A): number of elements in set A Finite Set: limited number of elements Infinite Set: unlimited number of elements Subset: set contained in another set

P is a subset of Q; {2, 3, 5} {1, 2, 3, 4, 5, 6} P is a Proper Subset of Q if but not equal to Union and Intersection Intersection: overlap, in both sets; and Union: everything in both sets; or Disjoint/Mutually Exclusive: no element in common

Example: P = {1, 3, 4} and Q = {2, 3, 5} Find Find Example M = {2, 3, 5, 7, 8, 9} and N = {3, 4, 6, 9, 10} True or False:

{9, 6, 3} N List the sets: Assignment P. 214 #1, 2, 4, 5-9 7B: Set Builder Notation TLW describe sets using set notation.

A={ Set of all x such that x is an integer between -2 and 4, including -2 and 4 A = {-2, -1, 0, 1, 2, 3, 4} n(A) = 7 means there are 7 elements in A How is B = {x| different from A? n(B) = ? Example

A = {x| List the elements of A n(A) = Try These P. 216 #1-4 7C: TLW identify the complement of a set. Universal Set: larger set we consider subsets of (U)

Complementary Set: U = {1, 2, 3, 4, 5} A = {2, 4} A = {1, 3, 5}; complement; elements in U but NOT in A Note: n(A) + n(A) = n(U)

Example Find C U={all positive integers} C={all even integers} Example U= C=

Example

U= A={x| B={ List the elements of the sets A B A B AB

Example Suppose U = {positive integers}, P= {multiples of 4 less

than 50} and Q = {multiples of 6 less than 50} List P and Q Find Find Verify that n() = n(P) +n(Q) n () Assignment P. 217 #1-13 odds 7D TLW diagram sets using Venn

Diagrams Venn Diagram: Universal set U (rectangle) sets inside are circles. U A A Subsets If , then everything in B is also in A.

U A B Intersection Elements in both A and B Overlap A

B Union All elements in A or B or both Disjoint or Mutually Exclusive No overlap Nothing in common

U A B Example Suppose U={1, 2, 3, 4, 5, 6, 7, 8}. Illustrate on a Venn diagram the sets: 1. A={1, 3, 6, 8} and B= {4, 5, 9} 2. A={1, 3, 6, 7, 8} and B={3, 6, 8}

Example Suppose U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. diagram the sets Illustrate on a Venn A={2, 4, 8} and B = {1, 3, 5, 9} Assignment P. 221 #1-7

7E TLW shade Venn diagram regions. Few can use shading to show various sets. U U A A is shaded U

A B is shaded B A B is shaded

B U A A is shaded B Example

Shade the following regions for two intersecting sets A and B: Assignment P. 222 #1-4 You can find PDFs of the Venn diagrams on my website. 8F TLW find numbers in regions.

Sets contained in a Venn diagram. U A B

Sometimes we want to know how many are in a region. U A How many elements are in the

universal set? B (7) (4) (6) (3)

Example How many elements are there in: P Q U P Q (7)

P, but not Q Q, but not P Neither P nor Q (4) (3)

(11) Things to Know Venn diagrams allow us to see identities. Assignment P. 224-225 #1-8

7G TLW problem solve with Venn diagrams. We can use Venn diagrams to help us solve problems. Example A squash club has 27 members. 19 have black hair, 14 have brown eyes, and 11 have both black hair and brown eyes. Places this information on a Venn diagram. Find the number of members with: black hair or brown eyes

black hair, but not brown eyes. Example A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dived from 5 m. How many dive from both platforms? Example A city has three football teams in the national league: A, B and C In the last season, 20% of the citys population saw

team A play, 24% saw team B and 28% say team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play. Create a Venn diagram and answer the following: 1. What percentage saw only team A play? 2. What percentage saw team A or team B play but not team C. 3. What percentage did not see any of the teams play?

Assignment P. 226-228 #1-8