Exam Review Suggestions	
•	Be able to tell if a set is well defined.
•	Be able to write a set in descriptive form, roster form, and set-builder form.
•	Be able to use the symbols ∈ and ∉ to show if an element is included in a set.
•	Be able to find the cardinality of a (finite) set.
•	Be able to use the symbols ⊆, ⊈, ⊂, and ⊄ to show if a set is a subset or proper subset of another set.
•	Be able to find the number of subsets or number of proper subsets that a given set has.
•	Be able to find the power set of a given set.
•	Be able to use Venn diagrams
•	Be able to find complements, unions, intersections, and relative complements of sets.
•	Be able to use a Venn diagram and given information to answer questions about the cardinality of sets in an application.

Practice Problems
1.  Determine if the following sets are well-defined.
a.	 The set of students in this class.
b.	 The set of natural numbers larger or equal to 10.
c.	 The set of old men living in the city of Saginaw.
d.	 The set of men born on or before December 31, 1929 living in the city of Saginaw.

2. Write the following set in roster form and in set builder form. The set of natural numbers larger or equal to 10.

3.	Let U = {red, orange, yellow, green, blue, indigo, violet}. Let A = {red, yellow, blue}, B = {green, blue, indigo, violet}, and C = {red, yellow, blue, green}.
Determine if the following is true or false.
a.	
a.	A  C
b.	A  C
c.	A   

d.	B – C = {red, yellow}
e.	B  C = {green, blue}
f.	n(P(B  C)) = 4
4. Draw a Venn diagram for each of these subsets of sets A, B, and C.
a.	(A – B)  (C – B)			b. ( B)  

5. Use Venn diagrams to determine the validity of the following statements.
a. A  (B  C) = (A  B)  C  		 
	             	    
b. A  (B  C) = (A  B)  (A  C)  

6. Let U = {x: x is a natural number, 1 < x < 15}, A = {1, 2, 3, 4, 5, 6, 7},  
	B = {3, 6, 9, 12, 15}, and C = {2, 4, 6, 8, 10, 12, 14}. Find the following.
a.	 A  (B  )

    b.	(A  B)  (C  B)

    c. 

   d. 

    e.	(A  B)  C 

   f. A  (B  C )

7. Out of 350 entering freshmen, 235 had to take a reading test, 210 had to take the writing test, and 245 had to take the math placement test. Of these students, 150 took all three tests, 170 took both reading and math placement, 155 took both writing and math, and 205 took both reading and writing. How many of the entering freshmen did not need to take any placement test?

8. Use Venn diagrams to show that (A  B) – C   (A – C)  B but that (A  B) – C  does not equal 
   (A – C)  B.

9.  Let U be the months of the year,  A = {January, March, May, July, August, October, December},  
B = {January, February, March, April, September, October, November, December}, C = {May, June, July, August}, and D = {January, April, June, September).  Find the following.

a.	(A  B)  (C   D)   

b.	[A  (B  C)]   D

c.	

d.	

e	

f.	P(B D)

g.	n(P(A))