CS 173 Homework 1 (due 9/5/08 in class) Fall 2008
CS 173: Discrete Mathematical Structures, Spring 2008
Homework 1
Due in class on Friday, September 5, 2008
Grading: The point values of the problems add up to 50 points.
1. [10 points] Translate the following sentences into propositional logic, making the meaning
of your propositional variables clear, and then create a truth table for each sentence. See
page 11 of the textbook for some examples of translating English sentences into propositional
logic.
(a) Either the Chicago White Sox pitching improves and they continue to hit well or the
Minnesota Twins will win the division.
(b) Discrete mathematics is interesting and has many useful applications or the students will
not be happy.
2. [4 points] Use a truth table to show that the following logical equivalence is correct
¬((p∧p)⊕q)≡(p↔q)
3. [10 points] In the following exercises, use the logical equivalences given on pages 24 and 25
of the textbook (in Tables 6 through 8) to show that:
(a) (¬p→(q→r)) ≡(q→(p∨r))
(b) ¬(p→ ¬q)∧ ¬(p∨q)is a contradiction (i.e. always false).
(c) (4 points) (p∨q)∧(¬p∨r)→(q∨r)is a tautology (i.e. always true)
4. [5 points] Assume that there are only two kinds of people, a person is either authentic or a
charlatan. A person is authentic if and only if every statement they make is true. A person is
a charlatan if and only if every statement they make is false. Suppose you meet Augustus De
Morgan and Charles Babbage in class one day and they say the following:
Babbage: Both De Morgan and I are authentic.
De Morgan: Babbage is a charlatan
What kind of people are De Morgan and Babbage? Justify your answer.
5. [5 points]
(a) State the negation of the statement “I have overslept or the building is on fire”, using
deMorgan’s laws to move the negation from the whole thing onto the two component
statements.
(b) Using your result from part (a), write the negation, contrapositive, converse and inverse
of the following statement (see page 8 of the textbook for a related example).
If I have overslept or the building is on fire, then the class will be canceled.
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