Worksheet on Logic
Benjamin Cosman, Patrick Lin and Mahesh Viswanathan
Fall 2020
Definitions from the Lecture
• A proposition is a statement that can either be true (de-
noted T) or false (denoted F).
• Propositions can be combined using logical operators not
(¬), or (∨), and (∧), implies (→), and if and only if (↔), to
create new propositions. Truth tables for these operations is
shown in Figures 1to 5.
• Two formulas/propositions Pand Qare logically equivalent
(denoted P≡Q) if they have the same meaning. That is,
Pand Qevaluate to the same value in all rows of a truth
table, or the formula P↔Qevaluated to Tin all rows of a
truth table.
• The contrapositive of an implication P→Qis the formula
(¬Q)→(¬P). The contrapositive (¬Q)→(¬P)is logically
equivalent to P→Q. The converse of an implication P→Q
is the formula Q→P.
• Predicates can either be universally quantified (∀x P(x)) or
existentially quantified (∃x P(x)) to create propositions from
predicates. Formulas can have multiple quantifiers and the
order in which they appear can influence their meaning.
• A domain of discourse identifies the set over which predi-
cate variables take values and the meaning of predicates. It
plays an important role in determining the truth of proposi-
tions.
P¬P
F T
T F
Figure 1: Truth table for ¬P
Problem 1.In which time zones are other students in your breakout
room?
Problem 2.Your class has a textbook and a final exam. Let P,Q, and
