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Let $U$ be the universe and $E\subsetneq U$ and $$S=\{x\in E \mid \exists V{\subseteq} E\;\; V{\in} \mathscr N(x)\}. \tag1$$

So, the complement of $S$ must contain elements outside $E,$ so I know that this negation isn't correct: $$S^\complement =\{x\in E \mid \forall V, V\cap E^c \neq \emptyset, V{\in} \mathscr N(x)\} .$$

I think that the correct negation is $$S^\complement= \{x\in U \mid \forall V, V\cap E^c \neq \emptyset, V{\in} \mathscr N(x)\},$$ but what are the coherent and systematic rules to apply to get here from $(1)$ ? I know that ∃ becomes ∀, etc, but what about the domain specification x ∈ E?

ryang
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niobium
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1 Answers1

2

  1. We negate a proposition or propositional function, and take the complement of a set.

  2. The negation of a tautology is a contradiction, whereas the complement of the set of tautologies contains contradictions as well as contingent sentences. So, negating a non-tautology doesn't generally give a tautology.

  3. $\\$

    \begin{align}\left\{x\in A \mid P(x)\right\}^\complement &=\left\{x\mid x\in A\:\:\land\:\:P(x)\right\}^\complement\\ &=\left\{x\mid \lnot\big(x\in A\:\:\land\:\:P(x)\big)\right\}\\ &=\left\{x\mid \big(x\not\in A\:\:\lor\:\:\lnot P(x)\big)\right\}\\ &=A^\complement\cup\left\{x\mid \lnot P(x) \right\}. \end{align}

    For example, in the universe $\mathbb Z,$ $$\left\{x\in \mathbb Z^+ \mid \exists k{\in}\mathbb Z\:\,x=7k\right\}^\complement=\mathbb Z_{\le0}\cup\left\{x\mid \forall k{\in}\mathbb Z\:\,x\ne7k\right\}.$$

ryang
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