Math 1090 Homework 1
Solutions
1. In translating each of the following to symbolic form, the word “or” would be replaced by ∨
or Y . What is the appropriate choice in each case? Explain why your choice is correct.
(a) To take discrete mathematics, you must have taken calculus or a course in computer
science.
Answer: ∨ since it is fine if you have taken both.
(b) Dinner for two includes two items from column A or three items from column B.
Answer: Y since if you choose two from A you cannot in addition choose three from B.
(c) When you buy a new car from Acme Motor Company, you get $ 2000 back in cash or a
2% car loan.
Answer: Y since you get one or the other but not both.
(d) School is closed if more than 2 feet of snow falls or if the wind chill is below –50.
Answer: ∨ since school is closed if both happen to be the case.
2. Determine whether
(∼ q ∧ (p → q)) → ∼ p
is a tautology.
Answer:
p
T
T
F
F
q
T
F
T
F
(∼ q
F
T
F
T
∧
F
F
F
T
(p → q))
T
F
T
T
→
T
T
T
T
∼p
F
F
T
T
Yes it is a tautology.
3. Does p → q logically imply (p ∧ q) ∨ ∼ p ? Justify your answer.
Answer:
p
T
T
F
F
q
T
F
T
F
p→q
T
F
T
T
(p ∧ q)
T
F
F
F
∨
T
F
T
T
∼p
F
F
T
T
Yes. Whenever p → q is T, so is (p ∧ q) ∨ ∼ p.
4. Use truth tables to show that both
p ∨ (p ∧ q) and p ∧ (p ∨ q)
are equivalent to p.
Answer:
p
T
T
F
F
q
T
F
T
F
p
T
T
F
F
∨
T
T
F
F
(p ∧ q)
T
F
F
F
p
T
T
F
F
∧
T
T
F
F
(p ∨ q)
T
T
T
F
1
5. Determine whether
(p ∨ q) → r and p ∨ (q → r)
are equivalent.
Answer: p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
(p ∨ q)
T
T
T
T
T
T
F
F
→
T
F
T
F
T
F
T
T
r
T
F
T
F
T
F
T
F
p
T
T
T
T
F
F
F
F
∨
T
T
T
T
T
F
T
T
(q → r)
T
F
T
T
T
F
T
T
Their truth tables are different. They are not equivalent.
6. (a) Determine whether
(p ↔ q) ↔ r and p ↔ (q ↔ r)
are equivalent.
(b) Is ((p ↔ q) ↔ r) ↔ (p ↔ (q ↔ r)) a tautology? Explain.
Answer:
(a) They are equivalent. Look at their truth tables:
p q r (p ↔ q) ↔ r p ↔ (q ↔ r)
T T T
T
T T T T
T
T T F
T
F F T F
F
T F T
F
F T T F
F
T F F
F
T F T T
T
F T T
F
F T F F
T
F T F
F
T F F T
F
F F T
T
T T F T
F
F F F
T
F F F F
T
(b) Look at the truth table above. ((p ↔ q) ↔ r) ↔ (p ↔ (q ↔ r)) is a tautology.
It always has truth value T.
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