CMSC 2123 – Discrete Structures
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Author 1:
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Author 2:
Student ID:
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Course:
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Assignment:
Due:
Scoring block
Exercise
Maximum
1
1
3
1
5
1
7
1
Total
4
p. 12 – 16
1,3,5,7
Ms. Fiona Faultless
*00000001
ffaultless@uco.edu
Ms. Petunia Perfect
*00000000
pperfect@uco.edu
CMSC 2123 – Discrete Structures
21389 Spring 2014
a01
January 15, 2014
Earned
1
1
1
1
4
Explanation
Assignment a01
Page 1 of 4
CMSC 2123 – Discrete Structures
p. 12 – 16
1,3,5,7
Assignment a01
Page 2 of 4
1. Which of these sentences are propositions? What are the truth values of those that are
propositions?
a) Boston is the capital of Massachusetts.
b) Miami is the capital of Florida.
c) 2 + 3 = 5
d) 5 + 7 = 10
e) 𝑥 + 2 = 11
f) Answer this question.
Answer:
Question
a
3.
a)
b)
c)
d)
Proposition
Yes
Truth Value
T
b
Yes
F
c
Yes
T
d
Yes
F
e
No
f
No
Explanation
The proposition is a statement of fact that can be
tested.
The proposition is a statement of fact that can be
tested.
The proposition is an arithmetic identity that can be
tested.
The proposition is an arithmetic identity that can be
tested.
The equation has infinitely many solutions that are
true and infinitely many solutions that are false. A
proposition can have only one truth value.
The proposition is a command and has no truth value.
What is the negation of each of these propositions?
Today is Thursday.
There is no pollution in New Jersey.
2 + 1 = 3.
The summer in Maine is hot and sunny.
Answer:
Question
a
b
c
d
Statement
Today is Thursday.
There is no pollution in New Jersey.
2 + 1 = 3.
The summer in Maine is hot and sunny.
Negation
Today is not Thursday.
There is pollution in New Jersey.
2 + 1 ≠ 3.
The summer in Maine is not hot or it is
not sunny.
CMSC 2123 – Discrete Structures
5.
p. 12 – 16
1,3,5,7
Assignment a01
Page 3 of 4
Let 𝑝 and 𝑞 be the propositions
𝑝: Swimming at the New Jersey shore is allowed.
𝑞: Sharks have been spotted near the shore.
Express each of these propositions as an English sentence.
a) ¬𝑞
b) 𝑝 ∧ 𝑞
c) ¬𝑝 ∨ 𝑞
d) 𝑝 → ¬𝑞
e) ¬𝑞 → 𝑝
f) ¬𝑝 → ¬𝑞
g) 𝑝 ↔ ¬𝑞
h) ¬𝑝 ∧ (𝑝 ∨ ¬𝑞)
Answer:
Question
a
b
Proposition
¬𝑞
𝑝∧𝑞
c
¬𝑝 ∨ 𝑞
d
𝑝 → ¬𝑞
e
¬𝑞 → 𝑝
f
¬𝑝 → ¬𝑞
g
𝑝 ↔ ¬𝑞
h
¬𝑝 ∧ (𝑝 ∨ ¬𝑞)
English Equivalent
Sharks have not been spotted near the shore.
Swimming at the New Jersey shore is allowed and sharks have been
spotted near the shore.
Swimming at the New Jersey shore is not allowed or Sharks have
been spotted near the shore.
If swimming at the New Jersey shore is allowed then Sharks have
not been spotted near the shore.
If sharks have not been spotted near the shore then swimming at
the New Jersey shore is allowed.
If swimming at the New Jersey shore is not allowed then sharks
have not been spotted near the shore.
Swimming at the New Jersey shore is allowed if and only if sharks
have not been spotted near the shore.
Swimming at the New Jersey shore is not allowed and, either
swimming at the New Jersey shore is allowed or sharks have not
been spotted near the shore.
CMSC 2123 – Discrete Structures
7.
p. 12 – 16
1,3,5,7
Assignment a01
Page 4 of 4
Let 𝑝 and 𝑞 be the propositions
𝑝: It is below freezing.
𝑞: It is snowing.
Write these propositions using 𝑝 and 𝑞 and logical connectives.
Answer:
Question
a
b
c
d
e
f
g
English Proposition
It is below freezing and snowing.
It is below freezing but not snowing.
It is not below freezing and not snowing.
It is either snowing or below freezing (or both).
If it is below freezing, it is also snowing.
If is either below freezing or it is snowing, but it is not
snowing if it is below freezing.
That it is below freezing is necessary and sufficient for it
to be snowing.
Mathematical Equivalent
𝑝∧𝑞
𝑝 ∧ ¬𝑞
¬𝑝 ∧ ¬𝑞
𝑞∨𝑝
𝑝→𝑞
𝑞 ∧ ¬𝑝
𝑞→𝑝