Reading Quiz Section 2.1
1. A tautology is a proposition which
(a) is false no matter what the truth value of its component propositions.
(b) is only true when all of its component propositions are true.
(c) is never false, no matter what the truth value of its component propositions.
(d) is built only using the connectives ∧, ∨.
2. A contradiction is a proposition which
(a) is false no matter what the truth value of its component propositions.
(b) is only true when all of its component propositions are false.
(c) is never false, no matter what the truth value of its component propositions.
(d) is built only using the connective ¬.
3. The contrapositive of the conditional P =⇒ Q is the conditional
(a) ¬P =⇒ Q
(b) ¬Q =⇒ ¬P
(c) Q =⇒ P
(d) P =⇒ ¬Q
4. True or False: The converse of P =⇒ Q is logically equivalent to P =⇒ Q.
5. The negation of the conditional “if I study at least 25 hours per week, then I will be successful”
is the proposition
(a) “I study at least 25 hours per week, but I am not successful.”
(b) “Either I study less than 25 hours per week, or I am successful.”
(c) “Either I study at least 25 hours per week, or I am not successful.”
(d) ‘If I am successful, then I will study at least 25 hours per week.”
6. De Morgan’s laws state that:
¬(P ∨ Q) is logically equivalent to (1)
¬(P ∧ Q) is logically equivalent to (2)
(a) (1) ¬(P =⇒ Q), (2) (¬P ∨ ¬Q)
(b) (1) (¬P ∧ ¬Q) , (2) (P ∨ Q)
(c) (1) (¬P ∧ Q) , (2) (P ∨ ¬Q)
(d) (1) (¬P ∧ ¬Q) , (2) (¬P ∨ ¬Q)
Practice Problems Section 2.1
1. Suppose that “If Colin was early, then no-one was playing pool” is a true statement.
(a) What is its contrapositive of this statement? Is it true?
(b) What is the converse? Is it true?
(c) What can we conclude (if anything?) if we discover each of the following? Treat the two
scenarios separately.
(i) Someone was playing pool.
(ii) Colin was late.
Video Solution
2. Prove that P ∨ ¬Q is logically equivalent to ¬P =⇒ (¬P ∧ ¬Q).
Video Solution
3. Define the connective ↑ (called the Sheffer stroke, or NAND) by the following truth table:
P Q P ↑ Q
T T F
T F T
F T T
F F T
(a) Prove P ↑ Q is logically equivalent to ¬(P ∧ Q).
(b) Find an expression built using only P and the connective ↑ which is logically equivalent
to ¬P.
(c) Find an expression built using only P, Q, and the connective ↑ which is logically equivalent
to P ∧ Q.
Video Solution
