Math 3345 Final with complete
verified solutions(graded A+)
Proposition - answer A statement that is true or false.
Logically equivalent - answer Describes sentences that have the
same truth values. Symbolized by "≡".
Logical connectives - answer Symbols that build compound
sentences out of simpler sentences. They include "not" or ¬, "and"
or ∧, "or" or ∨, "implies" or ⇒, and "if and only if" or ⇔.
Propositional calculus - answer The branch of logic concerned with
analyzing the truth values of compound sentences in terms of the
simpler sentences from which they are built.
Propositional variables - answer Letters like P, Q, R, etc. that
represent sentences.
Negation - answer In terms of P, this is ¬P.
Negative sentence - answer A sentence of the form ¬P. If a sentence
is of the form ¬P, it is called a negative sentence.
Conjunction - answer P∧Q. If a sentence is of the form P∧Q, it is
called a conjunctive sentence.
Commutative property - answer P∧Q ≡ Q∧P. Similarly, P∨Q ≡ Q∨P.
,Associative property - answer P∧(Q∧R) ≡ (P∧Q)∧R. Similarly, P∨(Q∨R)
≡ (P∨Q)∨R.
Conditional sentence - answer A sentence of the form P⇒Q, where P
is called the antecedent and Q is called the consequent.
Meanings of P⇒Q - answer "P implies Q", "If P, then Q", "P is
sufficient for Q", "Q is necessary for P", "Q if P", "Q is implied by P".
De Morgan's Laws - answer ¬(P∧Q) ≡ ¬P∨¬Q
¬(P∨Q) ≡ ¬P∧¬Q
Distributive Laws - answer P∧(Q∨R) ≡ (P∧Q)∨(P∧R)
P∨(Q∧R) ≡ (P∨Q)∧(P∨R)
Converse - answer In terms of P⇒Q, this is Q⇒P.
Contrapositive - answer In terms of P⇒Q, this is ¬Q⇒¬P. Note: P⇒Q ≡
¬Q⇒¬P.
Implication Laws - answer P⇒Q ≡ ¬P∨Q
¬(P⇒Q) ≡ P∧¬Q
Disjunction - answer P∨Q. If a sentence is of the form P∨Q, it is
called a disjunctive sentence.
Biconditional - answer P⇔Q. A sentence of the form P⇔Q is called a
biconditional sentence.
, Theorem (biconditional) - answer P⇔Q ≡ (P⇒Q)∧(Q⇒P). Therefore, "P
is necessary and sufficient for Q" means the same thing as "P if and
only if Q". P⇒Q is the forward implication and Q⇒P is the reverse
implication.
The order of priority of the logical connectives - answer ¬, ∧, ∨, ⇒,
⇔.
Tautology - answer A sentence that is true simply because of the
way it is built from ore basic sentences by means of the
connectives, and not because of the truth values of the basic
constituent sentences. The sentence must be true for all truth
assignments to its propositional variables.
Conditional proof - answer A method to prove A⇒B is true involving
merely considering the case where A is true and showing that in this
case, B must also be true.
Modus ponens - answer If P⇒Q is true and P is also true, then Q
must be true.
Contradiction - answer A sentence of the form Q∧¬Q. Such a
sentence is false regardless of the truth value of Q.
Proof by contradiction - answer A method of proving P that involves
assuming ¬P and deducing a contradiction Q∧¬Q.
Proof by contraposition - answer A method of proving P⇒Q merely
by proving ¬Q⇒¬P.
Three ways to prove an implication A⇒B - answer 1. Method of
conditional proof
2. Proof by contraposition
verified solutions(graded A+)
Proposition - answer A statement that is true or false.
Logically equivalent - answer Describes sentences that have the
same truth values. Symbolized by "≡".
Logical connectives - answer Symbols that build compound
sentences out of simpler sentences. They include "not" or ¬, "and"
or ∧, "or" or ∨, "implies" or ⇒, and "if and only if" or ⇔.
Propositional calculus - answer The branch of logic concerned with
analyzing the truth values of compound sentences in terms of the
simpler sentences from which they are built.
Propositional variables - answer Letters like P, Q, R, etc. that
represent sentences.
Negation - answer In terms of P, this is ¬P.
Negative sentence - answer A sentence of the form ¬P. If a sentence
is of the form ¬P, it is called a negative sentence.
Conjunction - answer P∧Q. If a sentence is of the form P∧Q, it is
called a conjunctive sentence.
Commutative property - answer P∧Q ≡ Q∧P. Similarly, P∨Q ≡ Q∨P.
,Associative property - answer P∧(Q∧R) ≡ (P∧Q)∧R. Similarly, P∨(Q∨R)
≡ (P∨Q)∨R.
Conditional sentence - answer A sentence of the form P⇒Q, where P
is called the antecedent and Q is called the consequent.
Meanings of P⇒Q - answer "P implies Q", "If P, then Q", "P is
sufficient for Q", "Q is necessary for P", "Q if P", "Q is implied by P".
De Morgan's Laws - answer ¬(P∧Q) ≡ ¬P∨¬Q
¬(P∨Q) ≡ ¬P∧¬Q
Distributive Laws - answer P∧(Q∨R) ≡ (P∧Q)∨(P∧R)
P∨(Q∧R) ≡ (P∨Q)∧(P∨R)
Converse - answer In terms of P⇒Q, this is Q⇒P.
Contrapositive - answer In terms of P⇒Q, this is ¬Q⇒¬P. Note: P⇒Q ≡
¬Q⇒¬P.
Implication Laws - answer P⇒Q ≡ ¬P∨Q
¬(P⇒Q) ≡ P∧¬Q
Disjunction - answer P∨Q. If a sentence is of the form P∨Q, it is
called a disjunctive sentence.
Biconditional - answer P⇔Q. A sentence of the form P⇔Q is called a
biconditional sentence.
, Theorem (biconditional) - answer P⇔Q ≡ (P⇒Q)∧(Q⇒P). Therefore, "P
is necessary and sufficient for Q" means the same thing as "P if and
only if Q". P⇒Q is the forward implication and Q⇒P is the reverse
implication.
The order of priority of the logical connectives - answer ¬, ∧, ∨, ⇒,
⇔.
Tautology - answer A sentence that is true simply because of the
way it is built from ore basic sentences by means of the
connectives, and not because of the truth values of the basic
constituent sentences. The sentence must be true for all truth
assignments to its propositional variables.
Conditional proof - answer A method to prove A⇒B is true involving
merely considering the case where A is true and showing that in this
case, B must also be true.
Modus ponens - answer If P⇒Q is true and P is also true, then Q
must be true.
Contradiction - answer A sentence of the form Q∧¬Q. Such a
sentence is false regardless of the truth value of Q.
Proof by contradiction - answer A method of proving P that involves
assuming ¬P and deducing a contradiction Q∧¬Q.
Proof by contraposition - answer A method of proving P⇒Q merely
by proving ¬Q⇒¬P.
Three ways to prove an implication A⇒B - answer 1. Method of
conditional proof
2. Proof by contraposition
