Q250, Homework 4: Propositional Logic

Q250, Homework 4:
Propositional Logic
5 October 2011
How to submit
Assignments must be emailed as a PDF attachment, to both myself (wrnthorn@indiana.edu) and
the UTI (jtorre@indiana.edu), with “Q250” in the subject line, no later than 9:30am on October
10th. The PDF file should be named q250 hw04 lastname.pdf where lastname is replaced by
your last name.
This and future assignments will involve lots of funny symbols. Most of the symbols for this
one have ASCII equivalents (like && for ∧, || for ∨, ∼ for ¬, => for ⇒, etc.). Thus it should be
easy to type up. However, you should explore your word processor’s support for these symbols and
for doing more complex layout, since we’ll need these in future assignments. Do not send scans
for this assignment unless you discuss it with me beforehand.
1
Let p be the proposition “you drive over 65 miles per hour” and let q be the proposition “you get
a speeding ticket”. Translate the following English sentences into propositions using p, q, and
logical connectives.
1. If you do not drive over 65 miles per hour, then you will not get a speeding ticket.
2. Driving over 65 miles per hours is sufficient for getting a speeding ticket.
3. Whenever you get a speeding ticket, you are driving over 65 miles per hour.
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Q250, Homework 4
2
Patricia, Quincy, and Rachel are in class together. Translate the following propositional statements
about them into the most natural equivalent English statements, where: p is the proposition “Patricia
will pass”, q is the proposition “Quincy will pass”, and r is the proposition “Rachel will
pass”.
1. ¬p∧¬q∧¬r
2. p∧q∧r
3. (p∧¬q∧¬r)∨(¬p∧q∧¬r)∨(¬p∧¬q∧r)
4. (q∨r)⇒¬p
3
State the converse and contrapositive for each of the following implications.
1. If it snows today, I will ski tomorrow.
2. I come to class whenever there is going to be a quiz.
4
Construct truth tables for each of the following propositions. Keep in mind that it may be useful
to record the truth values of intermediate expressions as well (e.g., when constructing a truth table
for (q∨r)⇒¬ j, it may be helpful to first write down the truth values for (q∨r)).
1. (p∨q)⇒(p∧q)
2. (p⇒q)⇔(¬q⇒¬p)
3. (p⊻q)∧(p⊻¬q)
4. (¬p⇔¬q)⇔(p⇔q)
5. (p⇔q)∨(¬q∧¬r)
6. (p⊻q)∨(q⇔r)
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Q250, Homework 4
5
Show that(p⇒q)⇒r and p⇒(q⇒r) are not equivalent.
6
For each of the following propositions, indicate whether it is a tautology, contradiction, or contingency.
If it is a contingency, provide two truth value assignments which demonstrate that this is
the case (i.e., provide one set of assignments that makes the proposition true and one that makes it
false).
1. ¬(p∨q)⇔(¬p∧¬q)
2. (p⇔q)∨(¬q∧¬p)
3. (p⇔q)∧(q⇒¬p)⇒(p∧r)
4. (p∧q)∧¬(p∨q)
5. ((p⇒q)∧(p⇒¬q))⇔¬p
7
Consider the following problem: Jones, Smith, and Clark hold the jobs of doctor, lawyer, and
veterinarian (not necessarily in that order). Jones owes the doctor $10. The veterinarian’s spouse
prohibits borrowing money. Smith is not married. Which person holds which job?
Use propositional logic to solve this problem. Use propositional symbols to represent the
nine different possible person/job assignments (e.g., JonesIsTheDoctor or just JD). Represent the
six possible solutions to this problem as logical expressions (e.g., one possible solution is JD∧
SL∧CV). Clues should be represented as additional logical expressions (e.g., from “Smith is not
married” and “the veterinarian’s spouse” you know that Smith cannot be the veterinarian, which
you can represent as¬SV). Show how the clues can be used to eliminate all but one of the possible
solutions.
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