# Project #42102 - Set Theory

Problem 0:  The rules for set theory have a duality. Show that the rules for arithmetic do not have a similar duality with even just plus and times. That is if you have a valid rule in arithmetic that involves multiplication and addition, then you cannot interchange the signs of multiplication and addition and obtain a valid rule. Give an example of this fact.

1. How many elements are in the union of four sets if each of the sets has 95 elements, each pair of sets share 46 elements, each triple of sets shares 26 elements and there are 4 elements in all four sets.

2. In a survey of 287 college students, it is found that 70 like brussels sprouts, 90 like broccoli, 57 like cauliflower, 30 like both brussels sprouts and broccoli, 23 like both brussels sprouts and cauliflower, 21 like both broccoli and cauliflower and 13 of the students like all three vegetables.
How many of the 287 college students do not like any of these three vegetables?

3. (2 pts) (a) If n ( A ) = 18, n ( B ) = 34 and n ( A ∩B ) = 5,
then n (A ∪B ) =

(b) If n ( A ∪B ) = 28, n ( A ∩B ) = 5 and n ( A ) = n ( B )
then n ( A ) =

4. (3 pts) Find the number of elements in A1 ∪A2 ∪A3 if there are 97 elements in A1, 1005 elements in A2 and 10014 elements in A3 in each of the following situations:
(a) The sets are pairwise disjoint.

(b) A1 ⊆ A2 ⊆ A3

(c) There are 15 elements common to each pair of sets and 5 elements in all three sets.

5. (4 pts) What is the cardinality of each of the following sets?

(a) ∅

(b) {∅}

(c) {∅, {∅}

(d) {∅, {∅}, {∅,{∅}}

6. (3pts) 167 business executives were surveyed to determine if they regularly read Fortune, Time, or Money magazines. 67 read Fortune, 70 read Time, 44 read Money, 42 read exactly two of the three magazines, 19 read Fortune and Time, 25 read Time and Money, and 2 read all three magazines.

How many read none of the three magazines?

How many read exactly one magazine?

How many did not read Money?

7.
A random sample of 260 students showed that 150 students are taking Math 245, and 200 are taking CS 108. How many are taking both?

8.
Suppose N(A) = 30, N(B) = 20, and N(A∩B) = 6
How many elements are in A∪B?

9.
Suppose N(A) = 100, N(B) = 200, N(C) = 300
N(A∩B) = 10, N(A∩C) = 15, N(B∩C) = 20
N(A∩B∩C) = 5
Therefore N(A∪B∪C) =

10.
A random sample of 300 people showed that 120 people like Italian food, 210 people like Mexican food, and 220 people like American food. If 60 like both Italian and Mexican, 75 like both Italian and American, and 100 like both Mexican and American, how many people like all three?

 Subject Mathematics Due By (Pacific Time) 10/05/2014 12:00 pm
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