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Brandon Rozek 2024-10-27 18:08:23 -04:00
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content/blog/implications-prenex-normal-form.md
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@ -37,11 +37,20 @@ $$
\begin{align*} \begin{align*}
(\forall x \phi) \implies \psi &\iff \neg (\forall x \phi) \vee \psi \tag{0.1} \\\\ (\forall x \phi) \implies \psi &\iff \neg (\forall x \phi) \vee \psi \tag{0.1} \\\\
&\iff (\exists x \neg \phi) \vee \psi \tag{2.2}\\\\ &\iff (\exists x \neg \phi) \vee \psi \tag{2.2}\\\\
&\iff \exists x (\neg \phi \vee \psi) \tag{2.1}\\\\ &\iff \exists x (\neg \phi \vee \psi) \tag{1.2}\\\\
&\iff \exists x (\phi \implies \psi) \tag{0.1} &\iff \exists x (\phi \implies \psi) \tag{0.1}
\end{align*} \end{align*}
$$ $$
**2.** Show that $\phi \implies (\exists x \psi)$ is equivalent to $\exists x (\phi \implies \psi)$ **2.** Show that $(\exists x \phi) \implies \psi$ is equivalent to $\forall x (\phi \implies \psi)$
$$
\begin{align*}
(\exists x \phi) \implies \psi &\iff \neg(\exists x \phi) \vee \psi \tag{0.1}\\\\
&\iff \forall x (\neg \phi) \vee \psi \tag{2.1}\\\\
&\iff \forall x (\neg \phi \vee \psi) \tag{1.1}\\\\
&\iff \forall x (\phi \implies \psi) \tag{0.1}
\end{align*}
$$
**3.** Show that $\phi \implies (\exists x \psi)$ is equivalent to $\exists x (\phi \implies \psi)$
$$ $$
\begin{align*} \begin{align*}
\phi \implies (\exists x \psi) &\iff \neg \phi \vee (\exists x \psi) \tag{0.1}\\\\ \phi \implies (\exists x \psi) &\iff \neg \phi \vee (\exists x \psi) \tag{0.1}\\\\
@ -51,3 +60,13 @@ $$
&\iff \exists x (\phi \implies \psi) \tag{0.1} &\iff \exists x (\phi \implies \psi) \tag{0.1}
\end{align*} \end{align*}
$$ $$
**4.** Show that $\phi \implies (\forall x \psi)$ is equivalent to $\forall x (\phi \implies \psi)$
$$
\begin{align*}
\phi \implies (\forall x \psi) &\iff \neg \phi \vee (\forall x \psi) \tag{0.1}\\\\
&\iff \forall x(\psi) \vee \neg \phi \tag{symmetry} \\\\
&\iff \forall x (\psi \vee \neg \phi) \tag{1.1}\\\\
&\iff \forall x (\neg \phi \vee \psi) \tag{symmetry} \\\\
&\iff \forall x (\phi \implies \psi) \tag{0.1}
\end{align*}
$$