Section 1.1 Propositional Logic 1

CHAPTER 1

The Foundations: Logic and Proofs

SECTION 1.1 Propositional Logic

2. Propositions must have clearly deﬁned truth values, so a proposition must be a declarative sentence with no

free variables.

a) This is not a proposition; it’s a command.

b) This is not a proposition; it’s a question.

c) This is a proposition that is false, as anyone who has been to Maine knows.

d) This is not a proposition; its truth value depends on the value of x.

e) This is a proposition that is false.

f) This is not a proposition; its truth value depends on the value of n.

4. a) Jennifer and Teja are not friends.

b) There are not 13 items in a baker’s dozen. (Alternatively: The number of items in a baker’s dozen is not

equal to 13.)

c) Abby sent fewer than 101 text messages yesterday. Alternatively, Abby sent at most 100 text messages

yesterday. Note: The ﬁrst printing of this edition incorrectly rendered this exercise with “every day” in

place of “yesterday.” That makes it a much harder problem, because the days are quantiﬁed, and quantiﬁed

propositions are not dealt with until a later section. It would be incorrect to say that the negation in that

case is “Abby sent at most 100 text messages every day.” Rather, a correct negation would be “There exists a

day on which Abby sent at most 100 text messages.” Saying “Abby did not send more than 100 text messages

every day” is somewhat ambiguous—do we mean ¬∀ or do we mean ∀¬?

d) 121 is not a perfect square.

6. a) True, because 288 > 256 and 288 > 128.

b) True, because C has 5 MP resolution compared to B’s 4 MP resolution. Note that only one of these

conditions needs to be met because of the word or .

c) False, because its resolution is not higher (all of the statements would have to be true for the conjunction

to be true).

d) False, because the hypothesis of this conditional statement is true and the conclusion is false.

e) False, because the ﬁrst part of this biconditional statement is false and the second part is true.

8. a) I did not buy a lottery ticket this week.

b) Either I bought a lottery ticket this week or [in the inclusive sense] I won the million dollar jackpot on

Friday.

c) If I bought a lottery ticket this week, then I won the million dollar jackpot on Friday.

d) I bought a lottery ticket this week and I won the million dollar jackpot on Friday.

e) I bought a lottery ticket this week if and only if I won the million dollar jackpot on Friday.

f) If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot on Friday.