Example: Let Example: Let U be the set of all senators in Congress. Then U = {x|x is a letter in the English alphabet} = {a, b, c, ..., z} A = {x|x is a vowel} = {a, e, i, o, u} D={x ∈ U| x is a Democrat} R={x ∈ U| x is a Republican} B = {x|x is a letter in the word texas} = { t, e, x, a, s} F={x ∈ U|x is a female} L={x ∈ U|x is a lawyer} Find the following sets and express your answer in a Venn diagram Find the region on the Venn diagram that corresponds to the set of b all senators that are democrats and are female or lawyers. Express c this set symbolically. z d A B y u a e i x f o t w g s v h j k l m n pq r a) What is AB?

b) What is Ac B?

c) What is A  Bc?

1 2 The Number of Elements in a Finite Set Example: A store has 150 clocks in stock. 100 of these clocks have AM or FM radios. 70 clocks had FM circuitry and 90 had AM The number of elements in set A is n(A). circuitry. How many had both AM and FM? How many were AM only? How many were FM only? if A = {x|x is a letter in the English alphabet}, then n(A)=26. Answer: Let If A = ∅ then n(A) = 0. U = {x|x is a clock in stock}, n(U) = A = {x|x is a set that can receive FM}, n(A) = B = {x|x is a set that can receive AM}, n(B) = We are also given that n(A B) = A B y Use the union rule to find the number in AB, x z

n(A B) =

UNION RULE: Find n(Ac B) = = how many have only AM.

n(ABc) = = how many have only FM.

3 4 Example: Harvesting Fruit - We are given the following data about Find the Venn diagram for the following the contents of some delivery trucks, One hundred shoppers are interviewed about the contents of their

34 trucks carried early peaches bags and the following results are found: 61 trucks carried late peaches 19 bought Twinkies 50 trucks carried extra late peaches 25 trucks carried early and late peaches 37 bought diet soda 30 trucks carried late and extra late peaches 18 bought broccoli 8 trucks carried early and extra late peaches 6 trucks carried all three kinds of peaches 1 bought broccoli, diet soda and Twinkies 9 trucks carried no peaches 11 bought Twinkies and diet soda

0 bought only Twinkies and broccoli

24 bought only diet soda

Now answer some questions:

How many carried only late peaches?

How many carried only one kind of peaches?

How many trucks went out?

5 6 Thirty-one children were asked about their lunch preferences and At a party, it was noticed that the following results were found: • 7 guests brought a salad, a dessert and drinks • 12 like cheeseburgers • 17 guests brought exactly two items • 14 like pizza • 23 guest brought drinks • 9 like burritos • 21 guests did not bring a salad • 5 like cheeseburgers and pizza • 9 guests brought only a dessert • 4 like cheeseburgers and burritos • 11 guests did not bring a drink or a dessert • 8 like pizza and burritos • 10 guests brought only a salad and a drink • 10 like none of these items • 11 guests brought a drink and a dessert How many guests were at the party? How many children like exactly one of these items?

7 8