Discrete Mathematics Answers

1. Suppose that A = {2, 4, 6, 8}, B = {2, 4, 8}, and C = {4, 8}. Write out the relationship between A and B, A and C, B and C.

2. Determine whether each of the following pairs of sets are equal.
a) {1, 3, 5, 7} and {3, 1, 7, 5}
b) {{1}} and {1, {1}}

3. Determine whether each of the following statement is true or false:
a) x ∈ {x}
b) {x} ⊆{x}
c) {x} ∈{x}
d) {x} ∈ {{x}}

4. Let A = {b, c, d, f, g}, B = {a, b, c}.
a) Find (A ⋃ B)
b) Find (A ⋂ B)
c) A – B
d) B – A

Solve this degeneracy problem
Dealer 1 2 3 4 Supply
A 2 2 2 4 1000
B 4 6 4 3 700
C 3 2 1 0 900
Demand 900 800 500 400

Solve this degeneracy problem
11 〖2 〗^8 8 〖6 〗^6 〖2 〗^4 Supply
〖9 〗^10 9 12 9 6 18
7 6 〖3 〗^8 7 7 10
〖9 〗^2 3 5 〖6 〗^2 11 8
Demand 12 8 8 8 4 4

Solve by NWCM (north-west corner method) and find optimal solution by UV method
3 1 7 4 Supply
2 6 5 9 250
8 3 3 2 350
Demand 200 300 350 400

Solve by least cost method and apply UV method to optimize the solution.
19 30 50 13 Supply
70 30 40 60 7
40 10 60 20 10
Demand 5 8 7 15 18

Solve the given problem by NWCM (north-west corner method), LCM (the least cost method) and VAM (Vogel's approximation method) and find optimal solution by UV method.
11 13 17 14 Supply
16 18 14 10 250
21 24 13 10 300
Demand 200 225 275 400

6. Show that ~ (p → q) and p ∧~q are logically equivalent. (Hint: you can use a truth table to prove it or you apply De Morgan law to show the ~(p → q) is p ∧~q.

7.Let p and q be the propositions.
p: I bought a lottery ticket this week.
q: I won the million-dollar jackpot on Friday.
a) Form a tautology using p. Express the tautology in English sentence.
b) Form a tautology using q. Express the tautology in English sentence.
c) Form a contradiction using p. Express the contradiction in English sentence.
d) Form a contradiction using q. Express the contradiction in English sentence.

8. If you have a tautology r and you negate r, what kind of sentence do you get?
a. A tautology
b. A contradiction
c. A sentence that is neither a contradiction nor a tautology
d. You can’t tell—it could be any of (a), (b), or (c).

5. Let p and q be the propositions.

p: You drive over 65 miles per hour.
q: You get a speeding ticket.

Write these propositions using p and q and logical connectives (including negations).

a) You do not drive over 65 miles per hour.
b) You drive over 65 miles per hour, but you do not get a speeding ticket.
c) You will get a speeding ticket if you drive over 65 miles per hour.
d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.
e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.
f) You get a speeding ticket, but you do not drive over 65 miles per hour.
g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

Answer each question completely, including all lettered parts.

1. Which of these sentences are propositions? What are the
truth values of those that are propositions?

a) Miami is the capital of Florida.
b) 2 + 3 = 5.
c) 5 + 7 = 10.
d) x + 2 = 11.
e) Answer this question.
f) What time is it?


2. What is the negation of each of these propositions?

a) Mei has an MP3 player.
b) There is no pollution in New Jersey.


3. Let p and q be the propositions.


p: I bought a lottery ticket this week.

q: I won the million-dollar jackpot on Friday.


Express of each of these propositions as an English sentence:

a) ~p
b) p^q
c) p v q
d) (~p) ^ (~q)


4. Let p, q, and r be the propositions

p: You have the flu.
q: You miss the final examination.
r: You pass the course.

Express each of these propositions as an English sentence.
a) p → q
b) ~q ↔ r
c) q →~r
d) p ∨ q ∨ r

Prove or disprove the following statement.
“Let p and q be positive integers, if p mod 4 = 1 and q mod 4 = 2, then pq mod 4 = 2.”