Math 2440

Content type
User Generated
Subject
Number Theory
School
CUNY Lehman College
Type
Homework
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1. No answer is required.
2.
a) ⏋𝑝: I didn’t buy a lottery ticket this week.
b) 𝑝 ⏋𝑞: I bought a lottery ticket this week but I didn’t win the million dollar jackpot.
c) 𝑞 𝑝: If I won the million dollar jackpot then I would have bought a lottery ticket this week.
d)
(
𝑝 𝑞
)
: It’s not the case that I bought a lottery ticket this week and I won the million dollar
jackpot.
e)
(
𝑝 𝑞
)
: It’s not the case that I bought a lottery ticket this week or I won the million dollar
jackpot.
3.
a) ⏋𝑝 𝑞: The election isn’t decided yet, however, the votes have been counted.
b)
(
𝑝 𝑞
)
: It’s not the case that the election is decided and the votes have been counted.
c) 𝑞 𝑝: For votes to have been counted it’s necessary that the election is decided.
d) 𝑟
(
𝑝 𝑞
)
: If Adam has been elected then the election is decided and the votes have been
counted.
e) (𝑟 𝑝) 𝑞: The votes have been counted and if Adam has been elected then the election is
decided.
4.
a) 𝑝 ⏋𝑞
(
𝑝 ⏋𝑞
)
.
b) ⏋𝑝 𝑞.
c) 𝑞 ⏋𝑝.
5.
a) 𝑟 ⏋𝑞.
b) 𝑟 𝑞.
c)
(
𝑞 𝑝
)
𝑟.
6.
a) 𝑞 ⏋𝑝.
b) (⏋𝑝 𝑞) 𝑟.
c) 𝑞
(
⏋𝑟 ⏋𝑝
)
.
7.
a) The truth table of 𝑝 𝑞:
𝑝
𝑞
𝑝 𝑞
T
T
T
T
F
F
F
T
F
F
F
F

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b) The truth table of 𝑞 𝑝:
𝑝
𝑞
𝑞 𝑝
T
T
T
T
F
F
F
T
F
F
F
F
It’s clear that conjunction is commutative since 𝑝 𝑞 and 𝑞 𝑝 have the same truth values.
c) The truth table of 𝑝 𝑞:
𝑝
𝑞
⏋𝑝
⏋𝑝 𝑞
T
T
F
F
T
F
F
F
F
T
T
T
F
F
T
F
d) The truth table of
(
𝑝 𝑞
)
:
𝑝
𝑞
⏋(𝑝 𝑞)
T
T
F
T
F
T
F
T
T
F
F
T
8.
a) The truth table of 𝑝 𝑞:
𝑝
𝑞
𝑝 𝑞
T
T
T
T
F
T
F
T
T
F
F
F
b) The truth table of 𝑞 𝑝:
𝑝
𝑞
𝑞 𝑝
T
T
T
T
F
T
F
T
T
F
F
F
It’s clear that disjunction is commutative since 𝑝 𝑞 and 𝑞 𝑝 have the same truth values.
c) The truth table of 𝑝 ⏋𝑞:
𝑝
𝑞
⏋𝑝
⏋𝑞
⏋𝑝 ⏋𝑞
T
T
F
F
F
T
F
F
T
T
F
T
T
F
T
F
F
T
T
T

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1. No answer is required. 2. ⏋𝑝: I didn’t buy a lottery ticket this week. 𝑝 ∧ ⏋𝑞: I bought a lottery ticket this week but I didn’t win the million dollar jackpot. 𝑞 → 𝑝: If I won the million dollar jackpot then I would have bought a lottery ticket this week. ⏋(𝑝 ∧ 𝑞): It’s not the case that I bought a lottery ticket this week and I won the million dollar jackpot. e) ⏋(𝑝 ∨ 𝑞): It’s not the case that I bought a lottery ticket this week or I won the million dollar jackpot. a) b) c) d) 3. ⏋𝑝 ∧ 𝑞: The election isn’t decided yet, however, the votes have been counted. ⏋(𝑝 ∧ 𝑞): It’s not the case that the election is decided and the votes have been counted. 𝑞 → 𝑝: For votes to have been counted it’s necessary that the election is decided. 𝑟 → (𝑝 ∧ 𝑞): If Adam has been elected then the election is decided and the votes have been counted. e) (𝑟 → 𝑝) ∧ 𝑞: The votes have been counted and if Adam has been elected then the election is decided. a) b) c) d) 4. a) 𝑝 ∨ ⏋𝑞 ∨ (𝑝 ∧ ⏋𝑞). b) ⏋𝑝 → 𝑞. c) 𝑞 → ⏋𝑝. 5. a) 𝑟 ∧ ⏋𝑞. b) 𝑟 → 𝑞. ...
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Anonymous
Awesome! Perfect study aid.

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