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Mathematics LibreTexts

2.S: Logical Reasoning (Summary)

  • Page ID
    7043
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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