Is this statement logically equivalent: ∃(() → ()) and ∀() → ∃()? Why? (I'm not sure how to proceed with this. I have to use the "universe of discourse" and not formal proof.)
Discrete math logic / proof question
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logic
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0Why do you have changed the question ? – 2017-02-11
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0I've rolled it back, If you need it, please ask a new question. – 2017-02-11
1 Answers
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$\exists x (P(x)\rightarrow Q(x))$ is equivalent to $\exists x (\neg P(x) \lor Q(x)),$ which is equivalent to $\exists x \neg P(x) \lor \exists x Q(x).$ This is equivalent to $\neg (\forall x P(x) \land \neg (\exists x Q(x)).$ From the definition of implication we then have the statement equivalent to $\forall x P(x) \rightarrow \exists x Q(x).$
I am not sure what it means to use a universe of discourse to provide a proof of logical equivalency. Perhaps this means to select a certain universe of discourse (which is simply a nonempty collection of objects) and then describe how each of the above statements are equivalent through informal appeal to examples from the universe of discourse.