EECS 210
Review Sheet for Test 2
Spring 2008
Given below are some questions to help get you started in your preparation for test 2. You
are responsible for all material covered in class even if there is no question below on that
topic.
1. Use Euclid’s algorithm to find gcd(700, 203)
700 = 3•203 + 91
203 = 2•91 + 21
91 = 4•21 + 7
21 = 3•7 + 0 Thus, the gcd is 7.
2. Let A = {1, 2, 3, w, z} and B = {1, {2}, {3}, {w, z}}. Find
a. B - A = { {2}, {3}, {w,z} }
b. n((B) = 16
c. List the subsets of A that are also subsets of B.
Ø and {1}
3. Let A be the set of English words that contain the letter x and let B be the set of English
words that contain the letter q. Express the following set as a combination of A and B.
The set of English words that contain neither x nor q. A'  B'
4. If A, B and C are nonempty sets, determine if each of the following is always, sometimes or
never true.
a. If A  B and B  C =  then A  C =  Always
b. A - (B  C) and B - (A  C) are disjoint Sometimes
c. If A  B = B, then A  B Sometimes
5. Shade a Venn diagram so the shaded area is the following set: (A - B)  (A - C)
6. Assume that both A and B are subsets of M. Consider the statement: If A  B, then
(M - B) (M - A). Prove the statement. Be sure to give reasons for each step.
Let x  M - B
x  M and x  B
xA
x  M- A
(M - B) (M - A)
Definition of difference
Given that A  B and definition of subset
Definition of difference
Definition of subset
7. If An = { x | x  N and x divides n with zero remainder}, find the elements of
a. A40 = {1, 2, 4, 5, 8, 10, 20, 40}
b. A0 = {1, 2, 3, …} all of Z+
8. For each of the following, determine if it is
A: a function that is one-to-one but not onto
B. a function that is onto but not one-to-one
C: a bijection
D: a function that is neither one-to-one nor onto
E: not a function
a. f: N  N defined by f(x) = x! D (Remember 0! = 1! = 1)
b. g: Z+  Z+ defined by g(x) = x/2 E (Not defined for x = 1)
c. h: Z  Z+ defined by h(x) = x2 + 1 D
9. Consider the function f: N  N defined by f(x) = x2
a. Prove or disprove that f is one-to-one.
Suppose f(x) = f(y)
thus, x2 = y2
so x = y since the domain is N, we do not need to say x =  y
Therefore, f is one-to-one
b. Prove or disprove that f is onto.
Suppose that f is onto and consider 2, an element of the codomain
if f(x) = 2, then x2 = 2 and thus x =2. Since 2 is not in N, 2 has no preimage
so f is not onto.
10. Consider the function g: Z  Z  Z defined by g(x, y) = x + y
a. Prove or disprove that g is one-to-one.
Since g((2, 3)) = g((1, 4)) = 5 g is not one-to-one
b. Prove or disprove that g is onto.
Let x be any integer in Z
Consider the ordered pair (x, 0)
Since g((x, 0)) = x, g is onto.
11. Let f and g be the function from Z to Z defined below:
f(x) = 2x2 + 3x
g(x) = 4x - 17
a. find gf
gf(x) = g(2x2 + 3x) = 4(2x2 + 3x) – 17 = 8x2 + 12x – 17
b. find fg
fg(x) = f(4x – 17) = 2(4x – 17)2 + 3(4x – 17) = 32x2 – 260 x + 527
11. Let A = {a, b, c} and let B = {a, c, d}. Define a relation R between (A) and (B) by
(C, D)  R if C  D. List all ordered pairs in R.
Subsets of A are
Subsets of B are

1. 
{a}
2. {a}
{b}
3. {c}
{c}
4. {d}
{a, b}
5. {a, c}
{a, c}
6. {a, d}
{b, c}
7. {c, d}
{a, b, c}
8. {a, c, d}
For ease of discussion the sets in (B) have been numbered. The following are the
subset relationships defined on the power sets:
 is a subset of all sets in (B) (8 ordered pairs)
{a} is a subset of 2, 5, 6, and 8 (4 ordered pairs
{b} is not the subset of any set in (B)
{c} is the subset of 3, 5, 7, 8 (4 ordered pairs)
{a, b} is not the subset of any set in (B)
{a, c} is the subset of 5 and 8 (2 ordered pairs)
{b, c} is not the subset of any set in (B)
{a, b, c} is not the subset of any set in (B)