EECS 210 Review Sheet
1. State the contrapositive of the following: If x or y is even then x•y is even
If xy is odd then x and y are odd
2. Write the statement below (in English) in the form if p then q
For n2 to be even, it is necessary that n is even.
If n2 is even then n is even
3. Consider the following propositions:
p: The food is good
q: The service is good
r: The rating is four-star
Write the following in symbolic form:
The food is good or the service is good, but not both.
pq
4. Consider the statement: For every x there is some y such that x2 + y2  0
a. Write the statement in symbolic notation: x y(x2 + y2  0)
b. Write the negation of the original statement (in English) below.
There is an x such that for every y, x2 + y2 < 0
5. Use a truth table to determine if the following pair of statements are equivalent.
A V (B  C) and (A V B)  (A  C)
A
T
T
T
T
F
F
F
F
B
T
T
F
F
T
T
F
F
C
T
F
T
F
T
F
T
F
B C AV (B  C) A V B
T
T
T
F
T
T
T
T
T
T
T
T
T
T
T
F
F
T
T
T
F
T
T
F
AC
T
F
T
F
F
F
F
F
(AVB)A C
T
F
T
F
F
F
T
T
They are not equivalent
6. 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
7. Shade a Venn diagram so the shaded area is the following set: (A - B)  (A - C)
8. 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
9. Let S denote the sum of three consecutive integers, the first of which is even.
Prove or disprove: S is not divisible by 6.
Let x (an even integer) be the first of the three integers, and let S denote their sum.
S=x+x+1+x+2
Algebra
= 3x + 3
Algebra
= 3(x + 1)
Algebra
Since x is even, x + 1 is odd.
Since x is the product of two odd integers, x is odd and thus not divisible by 6,
10. Prove that n! + 2 is even for all n ≥ 2.
Notice that since n  2 then n! has a factor of 2
definition of factorial
n! = 2k definition of factor
n! + 2 = 2k + 2 = 2(k + 1)
algebra
Therefore n! + 2 is even
definition of even
11. 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