1/19
2/17
3/10
4/15
5/11
6/7
7/7
8/11
Total/100
Please do not write in the spaces above.
Directions: You have 50 minutes in which to complete this exam. You must show all
work, or risk losing credit. Be sure to answer all questions asked.
Good luck! J
MATH 225
Name:
Spring 2015
Dr. Morton
Exam II
1.
(19 points) a. Symbolically write the converse of the implication A ⇒ B .
b. Symbolically write the contrapositive of the implication A ⇒ B .
c. Symbolically write the negation of the implication A ⇒ B .
d. Now consider the statement “If x is even and y is prime then z=x-y is non-zero.” Apply
parts a-c above and write the
§ converse of this statement (use positive language as much as possible)
§
contrapositive of this statement (use positive language as much as possible)
§
negation of this statement (use positive language as much as possible)
2. (17 points) Determine whether the following are true or false, giving an explanation as to
your work. Points will be awarded based on how convincing and clear your explanation is.
a. ∀x, y ∈, ∃z ∈ : (( x < z ) ∧ (z < y))
b. ∀x ∈, ∃z ∈ : ( z < x )
c. ∃z ∈ : ∀x ∈, ( z < x )
3. (10 points) For A = {a ∈ : a ≤ 1} and B = {b ∈ : b = 6}
a. List the elements of A.
b. List the elements of B.
c. List the elements of A × B .
d. List the elements of B × A .
e. Give a geometric description on the graph below of the points in the xy-plane
belonging to ( A × B ) ∪ ( B × A ) .
4. (15 points) Suppose we have the two open sentences
P(x) : x 2 = 4; Q(x) : x ≤ 0
over the domain S =  .
For which x ∈S are the following true? To get full credit, you must explain why it is true at the x’s
that you have listed, and why it is not true for the ones not listed.
a. P(x) ⇒ Q(x)
b. (  P(x)) ∧ Q(x)
5. (11 points) Carefully find the negation of the following statements. Write the negation as a
positive statement, to whatever extent is possible. (Points will be given for how well you
write the negation.)
a.
2 is rational or a is a negative integer
b. For every rational r there exists an irrational s such that r<s.
6. (7 points) Prove: Let n be an integer. If 1− n 2 > 2 then 3n − 2 is an even integer.
7. (7 points) Is the following true or false? Explain why, making sure that your explanation is
convincing.
For every prime number n, n+2 is prime.
8. (14 points)
a. Suppose Q(x,y) :=”x=y+3”. What is the truth value of Q(2,5) ?Why?
b. Show that the statement ( ( p) ∧ ( p ∨ q) ) ⇒ q is a tautology by filling in the following truth
table, being careful not to skip any needed columns.
p
q
T
T
T
F
F
T
F
F
How does your table tell you that it is a tautology?
c. Write the following statement as an “if-then” statement:
7 is odd if 15 is prime.
d. Is the following true or false? Why?
0 > 1 ⇔ 5 is even