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# 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