Name ______
MAT 101 – A Survey of Mathematical Reasoning March 6, 2014 Professor Pestieau Exam 1 – Set Theory
[Questions 1 – 8 are worth 5 points each]
True or False?
Determine whether the following statements are true (T) or false (F).
1. A set with cardinality 1 has two distinct proper subsets.
T F
2. No set can have exactly 14 distinct subsets.
T F
3. For any two finite sets A and B , n(A З B) < n(A И B).
T F
4. If A and B are both proper subsets of C , then the union of A and B is also a proper subset of C .
T F
5. The set {1,2,5} belongs to the power set of {1,2,3,4,5} .
T F Multiple-Choice Questions
Circle the correct answer for the following questions.
Let the universal set U be the set of letters in the word “uncopyrightable” (the longest word in the English language with no repeating letters).
Let A be the set of vowels in U (assume that “y” is not a vowel).
Let B= { a , c , o , p , t , u , y }.
6. Which of the following is the set A B ?
a) {a , o , u } b) {a , c , e , i , o , p , t , u , y }
c) {a , e , i , o , u } d) {a , b , c , e , g , h , i , l , n , o , p , r , t , u , y }
7. Which of the following is the set ( A B)/ ?
a) {b , g , h , l , n , r } b) {b , c , g , h , l , n , p , r , t , y }
c) {b , c , e , g , h , i , l , n , p , r , t , y } d) {a , c , e , i , o , p , t , u , y }
8. What is the cardinality of the set B- A/ ?
a) 3 b) 4
c) 11 d) 13 Show all your work on the following problems to receive full credit.
Problem 1
Consider the two sets A and B given below.
A = {x | x is an even natural with only one digit}.
B = {3, 4, 5, 6, 7, 8, 9, 12, 14, 15}.
a) Find the set A B . [5 pts]
b) List all the subsets of A B . [5 pts]
c) Let C be a set containing any 12 naturals. [Bonus – 5 pts]
What is the maximum possible cardinality of B C ? Give an example of a set C that would work in this case. Problem 2
a) Write a set expression for the shaded area in the Venn diagram above. [5 pts]
b) Write a second equivalent set expression for this shaded area. [Bonus – 5 pts]
Problem 3 [10 pts]
Suppose A is a set that consists of odd integers. If {- 1,5} and {- 3, - 1,7} are both proper subsets of A and A has exactly 16 distinct subsets, then what is A ? Explain your reasoning. Problem 4 [15 pts]
Show that, for any three sets A , B and C , the following set identity is always true:
C-( A� B ) 乔 C ( A B )/ .
Shade each side of the identity in the Venn diagrams below and use the numbering of the regions to justify your work. Survey Problem [20 pts]
The Financial Aid Director of a small community college surveyed the records of 100 sophomore students and came up with the following data:
49 students receive government grants. 55 students receive private aid. 43 students receive scholarships from the college. 23 students receive both government grants and private aid. 18 students receive both government grants and scholarships from the college. 28 students receive both private aid and scholarships from the college. 8 students receive help from all three sources.
Summarize the results of this survey in the Venn diagram given below and answer the given questions. How many students… a) …receive government grants only? ______b) …receive private aid but not government grants? ______c) …receive financial aid from only one of the three sources? ______d) …receive no aid from the college or from the government? ______e) …receive no financial aid from any of these sources? ______