2.S: Logical Reasoning (Summary)

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Important Definitions

Logically equivalent statements, page 43
Converse of a conditional statement, page 44
Contrapositive of a conditional statement, page 44
Equal sets, page 55
Variable, page 54
Universal set for a variable, page 54
Constant, page 54
Predicate, page 54
Open sentence, page 54
Truth set of a predicate, page 58
Universal quantifier, page 63
Existential quantifier, page 63
Empty set, page 60
Counterexample, page 66 and 69
Perfect square, page 70
Prime number, page 78
Composite number, page 78

Important Theorems and Results

Theorem 2.8. Important Logical Equivalencies. For statements \(P\), \(Q\), and \(R\),

De Morgan's Laws \(\urcorner (P \wedge Q) \equiv \urcorner P \vee \urcorner Q\)
\(\urcorner (P \vee Q) \equiv \urcorner P \wedge \urcorner Q\)

Conditional Statement \(P \to Q \equiv \urcorner Q \to \urcorner P\) (contrapostitive)
\(P \to Q \equiv \urcorner P \vee Q\)
\(\urcorner (P \to Q) \equiv P \wedge \urcorner Q\)

Biconditional Statement \((P \leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)\)

Double Negation \(\urcorner (\urcorner P) \equiv P\)

Distributive Laws \(P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)\)
\(P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)\)

Conditionals with Disjunctions \(P \to (Q \vee R) \equiv (P \wedge \urcorner Q) \to R\)
\(P \vee Q) \to R \equiv (P \to R) \wedge (Q \to R)\)

Theorem 2.16. Negations of Quantified Statements. For any predicate \(P(x)\),

\(\urcorner (\forall x) [P(x)] \equiv (\exists x) [\urcorner P(x)]\), and
\(\urcorner (\exists x) [P(x)] \equiv (\forall x) [\urcorner P(x)]\)

Important Set Theory Notation

Notation	Description	Page
\(y \in A\)	\(y\) is an element of the set \(A\).	55
\(z \notin A\)	\(z\) is not an element of the set \(A\).	55
{ }	The roster method	53
{\(x \in U \| P(x)\)}	Set builder notation	58