CmSc 180 Discrete mathematics
Homework 01: due 01/22
1. Let P and Q be the propositions:
P: You drive over 65 miles per hour
Q: You get a speeding ticket
Write the following propositions using P and Q and logical connectives:
a.
b.
c.
d.
e.
You do not drive over 65 miles per hour
You drive over 65 miles per hour, but you do not get a speeding ticket
You get a speeding ticket if you drive over 65 miles per hour
If you do not drive over 65 miles per hour then you will not get a speeding ticket
You get a speeding ticket but you do not drive over 65 miles per hour
2. Let P and Q be the propositions
P: I bought a lottery ticket this week
Q: I won the million dollar jackpot
A. Express each of the following propositions in English:
(time is not present in the logical expressions, so when translating you have to choose
appropriate verb tense, e.g. ~P  Q might be translated as: "If I don't buy a lottery
ticket, I will win the million dollar jackpot")
a.
b.
c.
d.
e.
~P
PQ
~P  ~Q
PQ
~P  ~Q
B. Construct the truth table of each expression above
P
T
T
F
F
Q
T
F
T
F
~P
~Q
PQ
~P  ~Q
PQ
~P  ~Q
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C. Write the negation of each expression in a) through e) above, and translate the
negation in English
Hints: Use De Morgan's laws, represent implication as a disjunction in order to apply
De Morgan's laws
Expression
~P
PQ
~P  ~Q
PQ
~P  ~Q
Negation
Translation
3. Consider the following statements:
a.
b.
c.
d.
e.
f.
g.
h.
If you miss the exam, then you fail the course
If you fail the course then you have missed the exam
If you don't miss the exam you don't fail the course
If you don't fail the course then you have not missed the exam
If you miss the exam then you don't fail the course
If you don't miss the exam then you fail the course
If you fail the course then you have not missed the exam
If you don't fail the course then you have missed the exam
A. Using P for "you miss the exam" and Q for "you fail the course", represent each
statement as a logical expression
B. For each statement in a) through h) indicate its contrapositive, its converse and its
inverse. Write your findings in a table like this:
Statement
a. P  Q
b. Q  P
c. ~P  ~Q
d. ~Q  ~P
e. P  ~Q
f. ~P  Q
g. Q  ~P
h. ~Q  P
Contrapositive
~Q  ~P (d)
Converse
Q  P (b)
Inverse
~P  ~Q (c)
C. There are four pairs of equivalent statements. Which are they?
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4. Consider the statement: Tax rates will be reduced if Anita wins the election
a.
b.
c.
d.
Identify atomic statements and represent the statement as a logical expression
Rephrase the sentence as “unless” sentence
Rephrase the sentence as ……… is sufficient for …………
Rewrite the logical expression as a disjunction and translate it in English
5. Consider the statement: A square has four equal sides
a. Rephrase the sentence as “if … then…” sentence, identify atomic statements and
represent the statement as a logical expression
b. Rephrase the sentence as “unless” sentence
c. Rephrase the sentence as “only if” sentence
d. Rephrase the sentence as ……… is sufficient for …………
e. Rephrase the sentence as ……… is necessary for …………
f. Write the contrapositive expression and translate it in English
g. Rewrite the logical expression as a disjunction and translate it in English
6. Using the equivalence laws show that (A V B)  (A V ~B)  A
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