Sample True or False Questions
Answers: 1F, 2F, 3F, 4T, 5T, 6F, 7T
Decide whether the statements below are TRUE or FALSE.
1.
The empty set is a proper subset of every set.
Not true because the empty set is not a proper subset of itself!
2.
If A  1, 2,3 , then 2  PA , where PA is the power set of A.
Not true because the elements of the power set of A are sets themselves
(subsets of A) and 2 is just a number.
3.
For any two finite sets A and B , if A  B , then n( A)  n( B) .
Not true when A=B.
4.
For any two sets A and B , the intersection of A and B is always a subset of A .
True. Check this with Venn diagrams (and don’t forget to include the special
cases with A=B and with A and B disjoint)
5.
A set with 4 elements has exactly 10 proper subsets with cardinality greater than 1.
True. Out of the 15 proper subsets the set has here, 5 have to be discarded
because their cardinality is 1 or less: the empty set and all 4 sets which have
only one element. That leaves out exactly 10 proper subsets with cardinality
greater than 1.
6.
The set of large naturals with 100 digits or more is well-defined.
False. It is not clear here what exactly constitutes a “large” natural with 100
digits… So the set is not weel-defined.
7.
For any two finite sets A and B , n( A  B)  n( A) only if B is a subset of A .
True. Again, check this with Venn diagrams considering all cases.
Sample Multiple-Choice Questions
Answers: 1B, 2D
  {1, 2, 3, 4, …} is the set of naturals, or counting numbers.
Consider the universal set U and the sets A and B defined below.
U  {x | x   and 2  x  18 } = {2, 3, 4, …, 17, 18}
A = {2x | x = 2, 3, …, 7} = {4, 6, 8, 10, 12, 14}
B = {3, 4, 5, 6, 7, 9, 12, 14, 15}
1.
Which of the following is the set A  B ?
a)
c)
2.
{3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15}
{4, 6, 8, 10, 12, 14}
b)
d)
{4, 6, 12, 14}
{3, 4, 5, 6, 7}
b)
d)
{2, 11, 13}
{2, 11, 13, 16, 17, 18}
Which of the following is the set  A  B  ?
/
a)
c)
{3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15}
{2, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18}
Sample Problem 1
Let U  a, b, c, d , e, f , g, h be the universal set and A, B, C be subsets of U defined as
follows:
A  a, b, c, d  ,
B  b, e, a, d  ,
C  a, e.
Find the following:
a)
A/  C
b)
A B C/
Answer:
{a, e, f, g, h}

Answer:
{b, d}
c)
 A  C /  B
Answer:
{c, f, g, h}
d)
All proper subsets of A/
Answer:
{ }, {e}, {f}, {g}, {h}, {e, f}, {e, g}, {e, h},
{f, g}, {f, h}, {g, h}, {e, f, g}, {e, f, h},
{e, g, h}, {f, g, h}

Sample Problem 2
Use the Venn diagram below to shade the region corresponding to the set
 X  Y    X  Z   Y  Z  .
For your convenience the 8 regions in the diagram have been numbered arbitrarily.
Answer
Looking at the numbered regions in the Venn Diagram above, we have:
X – Y = {2, 3, 4, 5} – {3, 4, 7, 8} = {2, 5}
X – Z = {2, 3, 4, 5} – {4, 5, 6, 7} = {2, 3}
Y – Z = {3, 4, 7, 8} – {4, 5, 6, 7} = {3, 8}
Therefore,
 X  Y    X  Z   Y  Z  = {2, 3, 5, 8}
Regions 2, 3, 5, 8 should be shaded in the Venn diagram above.
Sample Problem 3
a)
Write a set expression for the region shaded in the Venn diagram above using any set
operations.
Here are two possible answers:
b)
 A  B    B  A
 A  B   A  B
Write an equivalent expression for the same region using all four set operations. [Bonus]
Here is one possible answer:
 A  B   A  B
/
/
Sample Problem 4
Show the set identity A  ( A/  B)  A  B using Venn diagrams.
Here you must show that the two shaded Venn diagrams associated with each side
of this set expression are identical. Use a numbering of the 4 regions for the lefthand side, and then check that you have shaded the union of A and B.
Sample Problem 5
Show that, in general, A  ( B  C )  ( A  B)  C using Venn diagrams.
Here you must show that the two shaded Venn diagrams associated with each side
of this set expression are not the same. Use a numbering of the 8 regions to get
these shadings.
Sample Problem 6
Let U, the universal set, consist of all the positive rational numbers (or fractions).
p
Let A be the set defined by A   | p, q are odd naturals and q  0 }. Find a set B that satisfied
q
the following 3 conditions:
1)
B is a subset of the complement of A.
2)
B has exactly 255 proper subsets.
3)
No element of B is greater than 1.
Here are two possible answers (out of many):
B = {1/2, 1/4, 1/6, 1/8, 1/10, 1/12, 1/14, 1/16}
or
B = {1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9}
Sample Survey Problem 1
Exercise # 17 p. 93 [Investing]
Answers to this problem are in the back of the book.
Sample Survey Problem 2
Exercise # 23 p. 93 [Gun Ownership]
Answers to this problem are in the back of the book.
Sample Survey Problem 3
Exercise # 25 p. 94 [Diets]
Answers to this problem are in the back of the book.