I need to understand the proof of $\sim\!\!(\forall x)A(x)\!\iff\!(\exists x)\sim \!\! A(x)$ in "A Transition to Advanced Mathematics: Edition 8" pg. 23 Theorem 1.3.1 a)
It states:
- Let $U$ be any universe (universe of discourse, the set of values being considered for $x$)
- The sentence $\sim\!\!(\forall x)A(x)$ is true in U
- iff the truth set of $A(x)$ is not the universe
- iff the truth set of $\sim A(x)$ is nonempty
- iff $(\exists x)\sim \!\! A(x)$ is true in U
I couldn't rationalize how "iff the truth set of $\sim A(x)$ is nonempty" led to " iff $(\exists x)\sim \!\! A(x)$ is true in U". Couldn't we also say "iff $(\forall x)\sim \!\! A(x)$ is true in U"?
Is there a better proof I can understand? So far I learned about negation, conjuctions, disjunctions, conditionals and biconditionals.
\lnot
for $\lnot$. $\endgroup$ – Dave Jan 20 at 16:43