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    Boolean Algebra

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    Boolean Algebra Homework Help Required

    Translating formal propositions
    ¬ = not ∃ =there exists ∨ = or ≡ equivalent
    ∀= for all ⇒ =implies ∧ = and ⇔ if and only if

    (a)An argument to show that two propositions are equivalent using the rules of Boolean algebra is given below. Give a rule or rules of Boolean algebra which can be used to justify each of steps (2) and (3) in this argument.

    ¬∀x in X[p(x) ⇒q(x)]
    ≡¬∀x in X[¬p(x) ∨q(x)] (rewriting ⇒)
    ≡∃x in X[¬(¬p(x) ∨q(x))] (2)
    ≡∃x in X[p(x) ∧¬q(x)] (3)

    (b)Using the rules of Boolean algebra show the following. State which rule is used at each step.

    (ii) ¬∃x in X[p(x) ∧¬q(x)] ≡∀x in X[¬p(x) ∨q(x)]

    (c) Part of a formal argument is given below. This argument has premises

    hot ⇒(cloudy ∨¬wet )
    hot ∨cloudy
    ¬cloudy

    and its aim is to deduce the conclusion ¬wet.

    Argument

    hot ⇒(cloudy ∨¬wet ) (1), premise
    hot ∨cloudy (2), premise
    ¬cloudy (3), premise
    hot (4), ?? 3, 2
    cloudy ∨¬wet (5), ?? 4, 1

    (i) For each of steps (4) and (5) in the argument above, say which rule of inference has been used.

    (ii) Using the rules of inference, complete a formal argument to reach the conclusion ¬wet .

    For easy reference purposes the rules of inference are: Implication; Contraposition; Equivalence; Disjunction; Combining ∨; Transitivity; Addition, Conjunction; Simplification.

    For (a) & (b) above the steps presumably relates to De Morgan rules of associativity, negation and absorption.

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    Boolean Algebra Homework Help Required

    Translating formal propositions
    ¬ = not ∃ =there exists ∨ = or ≡ equivalent
    ∀= for all ⇒ =implies ∧ = and ⇔ if and only if

    (a)An argument to show that two propositions are equivalent using the rules of Boolean algebra is given below. Give a rule or rules of Boolean algebra which can be used to justify each of steps (2) and (3) in this argument.

    ¬∀x in X[p(x) ⇒q(x)]
    ≡¬∀x in X[¬p(x) ...

    $2.19