How to convert the following statement with the existential quantifier to statement with universal quantifier?
$\exists n. n>1\rightarrow x(n)\not=1$
Please give me some suggestion?
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1$\begingroup$ It is not clear what you mean by "with universal quantifier". Generally you can not convert a "universal formula" to an "existential formula" (there are no $P(x)$ and $Q(x)$ where $\exists x; Q(x) \equiv \forall x; P(x)$ in the same semantics). But I guess this will help you, a basic property of quantifiers which asserts that the negation of some universally quantified formula equals to the existential quantification of its negation. Then you just need a double negation. $\endgroup$– BelegCommented Jul 4, 2018 at 8:39
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Let's give names for the atom formulas:
$n>1$ will be $\phi$ and $x(n) \neq 1$ will be $\psi$ Now what we have is: $$\exists{n} (\phi\rightarrow\psi)$$ $$\exists{n} (\neg\phi \lor\psi)$$ $$\neg\neg \exists{n} (\neg\phi \lor\psi)$$ $$\neg\forall{n} \neg(\neg\phi \lor\psi)$$ $$\neg\forall{n} (\phi \land \neg\psi)$$ $$\neg\forall{n}((n>1) \land \neg(x(n)\neq 1))$$ $$\neg\forall{n}((n>1) \land (x(n) = 1))$$
answered Jul 4, 2018 at 22:04