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Math Help - Logical Equivalences with quantifiers

  1. #1
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    Logical Equivalences with quantifiers

    Hello

    I am running into problems with the following:

    x(P (x) ⇒ (Q(x) R(x)))⇐⇒ (xP (x) ∨ ∀yQ(y) ∨ ∃zR(z))

    Although I do not master logical equivalences, I have been able to solve some earlier. However, since this problem has quantifiers, I don't know how to approach it.

    Any help is highly appreciated.
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  2. #2
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    Re: Logical Equivalences with quantifiers

    P(x) => (Q(x) => R(x)) is equivalent to ~P(x) \/ ~Q(x) \/ R(x).

    Existential quantifier distributes over disjunction (because existential quantifier is basically a disjunction over all elements of the domain): \exists x\,(A(x)\lor B(x))\iff(\exists x\,A(x))\lor(\exists x\,B(x)).

    Next, existential quantifier changes into universal quantifier and vice versa when it is moved through a negation: \exists x\,(\neg A(x))\iff\neg(\forall x\,A(x)).

    These equivalences are sufficient to derive the one you need.
    Thanks from aprilrocks92
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