Truth Values
Equivalences
Equivalence Laws
Predicates
Quantifiers
Scope, Negation, & Restricted Domains
100
What do we call a compound proposition that is true under every truth assignment in a truth table?
A tautology.
Example: p ∨ ¬p.
100
Complete the equivalence: p → q ≡ _____.
¬p ∨ q
Conditional–disjunction equivalence.
OR
¬q → ¬p
Contrapositive
100
Simplify ¬(p ∧ q). What is the name of the Equivalence Law?
¬p ∨ ¬q
De Morgan’s Law changes AND to OR and negates each part.
100
Let P(x) mean “x > 3.” Is P(4) true or false?
True.
Substitution makes the predicate a proposition: 4 > 3.
100
Which symbol means “for every x”?
∀x.
The Universal Quantifier.
100
Negate ∀x P(x).
∃x ¬P(x).
A universal claim fails when a counterexample exists.
200
Classify p ∧ ¬p. Is it a
a) Contingency
b) Tautology
c) Contradiction
A contradiction.
It is false whether p is true or false.
200
Which form is equivalent to p → q: its converse or its contrapositive?
Its contrapositive: ¬q → ¬p.
The converse q → p does not have the same truth values.
200
Simplify ¬(p ∨ q). What is the name of the Equivalence Law?
¬p ∧ ¬q.
De Morgan’s Law changes OR to AND.
200
If the domain is integers, is P(pi) defined if P(x) means “x > 0”?
No. 
is outside the domain.
The input must belong to the predicate’s domain.
200
Which symbol means “there exists an x”?
∃x.
The Existential Quantifier.
200
Negate ∃x P(x).
∀x ¬P(x).
No value exists for P.
300
Classify p ∧ q. Is it a
a) tautology,
b) contradiction
c) contingency
A contingency.
It is true when both variables are true and false for other assignments.
300
When are two compound propositions logically equivalent?
When their truth values agree for every assignment.
Equivalently, their biconditional is a tautology.
300
Simplify p ∨ (p ∧ q). What is the name of the Equivalence Law?
p.
Absorption Law.
300
Let Q(x,y) mean “x = y + 3.” Evaluate Q(3,0). Is it true or false?
True.
3 = 0 + 3.
300
Given that the domain is integers, is ∀x (x² > 0) true or false? Give a reason.
False. x = 0 is a counterexample.
One counterexample disproves a universal statement.
300
If the domain is all students at Temple University, translate “Every CIS 1166 student studies logic.” Let C(x) mean “x takes CIS 1166” and L(x) mean “x studies logic.”
∀x (C(x) → L(x)).
A restricted domain uses a conditional.
400
Classify (p ∧ q) → p Is it a?
a) contradiction
b) contingency
c) tautology
It is a tautology.
When the hypothesis (p ∧ q) is true, p is true. Otherwise the conditional is true.
400
Rewrite ¬(p → q) without using a conditional statement.
p ∧ ¬q
Negate ¬p ∨ q using De Morgan’s law.
400
Simplify p ∧ T and p ∨ F. What is the name of the Equivalence Law?
Both simplify to p.
These are Identity Laws.
400
Why is “x < 100” by itself not a proposition?
Its truth value depends on x which is undefined.
A value for x or a quantifier can make a truth-valued statement.
400
Over the positive integers, is ∃!x (x2 = 4) true or false?
Yes. x = 2 is the only solution.
The negative solution lies outside the domain.
400
If the domain is all students at Temple University, translate “Some CIS 1166 student studies logic” using C(x) and L(x).
∃x (C(x) ∧ L(x)).
A restricted existential claim uses conjunction.
500
Give a truth assignment for p and q that makes the following expression true.
(p ∨ q) ∧ ¬p
p = F and q = T.
The first part needs q to be true while ¬p requires p false.
500
Give one assignment for p, q, and r that satisfies the expression
(p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r)
For example, p = T, q = F, r = F.
At least one variable must be true and at least one false.
500
Simplify p ∨ (q ∧ r). What is the name of the Equivalence Law?
(p ∨ q) ∧ (p ∨ r).
Distributive Law.
500
Let R(x,y,z) mean “x + y = z.” Evaluate R(0,0,1). Is it true or false?
False.
0 + 0 ≠ 1.
500
Given the finite domain {1,2,3}, expand ∃x P(x) using only P(1), P(2), P(3).
P(1) ∨ P(2) ∨ P(3).
Existential quantification over a finite domain acts like OR.
500
In ∃x (x + y = 1), which variable is not bound?
y is not bound; x is bound by ∃x.
The quantifier ∃x applies to x within its scope.