Chapter 1 Flashcards
Truth Table for Negation of Proposition:
when P is TRUE,
¬P is ___
FALSE
Truth Table for Negation of Proposition:
when P is FALSE,
¬P is ___
TRUE
The Truth Table for the Conjunction of Two Propositions:
When P is TRUE & Q is TRUE,
P ∧ Q is ___
TRUE
The Truth Table for the Conjunction of Two Propositions:
When P is TRUE & Q is FALSE,
P ∧ Q is ___
FALSE
The Truth Table for the Conjunction of Two Propositions:
When P is FALSE & Q is TRUE,
P ∧ Q is ___
FALSE
The Truth Table for the Conjunction of Two Propositions:
When P is FALSE & Q is FALSE,
P ∧ Q is ___
FALSE
The Truth Table for the DISJUNCTION of Two Propositions:
When P is TRUE & Q is TRUE,
P ∨ Q is ___
TRUE
The Truth Table for the DISJUNCTION of Two Propositions:
When P is TRUE & Q is FALSE,
P ∨ Q is ___
TRUE
The Truth Table for the DISJUNCTION of Two Propositions:
When P is FALSE & Q is TRUE,
P ∨ Q is ___
TRUE
The Truth Table for the DISJUNCTION of Two Propositions:
When P is FALSE & Q is FALSE,
P ∨ Q is ___
FALSE
What is:
P ⊕ Q
EXCLUSIVE OR;
The “exclusive or” of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true and is false otherwise.
The Truth Table for the Exclusive Or of Two Propositions:
When P is TRUE & Q is TRUE,
P ⊕ Q is ___
FALSE
The Truth Table for the Exclusive Or of Two Propositions:
When P is TRUE & Q is FALSE,
P ⊕ Q is ___
TRUE
The Truth Table for the Exclusive Or of Two Propositions:
When P is FALSE & Q is TRUE,
P ⊕ Q is ___
TRUE
The Truth Table for the Exclusive Or of Two Propositions:
When P is FALSE & Q is FALSE,
P ⊕ Q is ___
FALSE
In P → Q, what is the HYPOTHESIS?
P
also called the ANTECEDENT or PREMISE
In P → Q, what is the CONCLUSION?
Q
also called the CONSEQUENCE
What are two other ways to say In P → Q, in English other than
“If P, then Q”
P only if Q;
Q unless ¬P;
The Truth Table for the Conditional Statement P → Q.
When P is TRUE, & Q is TRUE,
P → Q is ___
TRUE
The Truth Table for the Conditional Statement P → Q.
When P is TRUE, & Q is FALSE,
P → Q is ___
FALSE
The Truth Table for the Conditional Statement P → Q.
When P is FALSE, & Q is TRUE,
P → Q is ___
TRUE
The Truth Table for the Conditional Statement P → Q.
When P is FALSE, & Q is FALSE,
P → Q is ___
TRUE
What is the CONVERSE of P → Q?
Q → P
What is the CONTRAPOSITIVE of P → Q?
¬Q → ¬P
What is the INVERSE of P → Q?
¬P → ¬Q
Which of the following has the same truth statements as P → Q:
CONVERSE, CONTRAPOSITIVE, or INVERSE?
CONTRAPOSITIVE
What does it mean for two propositions to be EQUIVALENT?
They have the same truth values
What are the contrapositive, the converse, and the inverse of the conditional statement
“The home team wins whenever it is raining?”
CONTRAPOSITIVE: “If the home team does not win, then it is not raining.”
CONVERSE: “If the home team wins, then it is raining.”
INVERSE: “If it is not raining, then the home team does not win.”
What is:
P ↔ Q
P if and only if Q (biconditional)
The biconditional statement P ↔ Q is true when p and q have the same truth values, and is false otherwise. Biconditional statements are also called bi-implications.
Truth Table for BICONDITIONAL P ↔ Q:
if P is TRUE, & Q IS TRUE,
P ↔ Q is ___
TRUE
Truth Table for BICONDITIONAL P ↔ Q:
if P is TRUE, & Q IS FALSE,
P ↔ Q is ___
FALSE
Truth Table for BICONDITIONAL P ↔ Q:
if P is FALSE, & Q IS TRUE,
P ↔ Q is ___
FALSE
Truth Table for BICONDITIONAL P ↔ Q:
if P is FALSE, & Q IS FALSE,
P ↔ Q is ___
TRUE
What is a tautology?
a compound proposition that is always true no matter what the truth values of the variables in it are.
What does it mean to be logically equivalent?
When compound propositions that have the same truth values in all possible cases are called logically equivalent
this also means that if p ↔ q is a tautology, then p and q are logically equivalent p ≡ q
What is a contradiction?
A compound proposition that is always false
What is a contingency?
A compound proposition that is neither a tautology nor a contradiction
what is
∀xP(x)?
universal quantification of P(x), read “for all xP(x)” or “for every xP(x).”
what is a COUNTEREXAMPLE
an element for which P(x) is false
Let Q(x) be the statement “x < 2.” What is the truth value of the quantification ∀xQ(x), where the domain consists of all real numbers?
Q(x) is not true for every real number x, because, for instance, Q(3) is false. That is, x = 3 is a counterexample for the statement ∀xQ(x).
Thus ∀xQ(x) is false.
what is
∃xP(x)
existential quantification
“There exists an element x in the domain such that P(x).”
