Non-Equivalence

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Theorem

Non-Equivalence as Disjunction of Conjunctions

Formulation 1

$\neg \left ({p \iff q}\right) \dashv \vdash \left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)$

Formulation 2

$\vdash \left({\neg \left ({p \iff q}\right)}\right) \iff \left({\left({\neg p \land q}\right) \lor \left({p \land \neg q}\right)}\right)$


Non-Equivalence as Disjunction of Negated Implications

$\neg \left ({p \iff q}\right) \dashv \vdash \neg \left({p \implies q}\right) \lor \neg \left({q \implies p}\right)$


Non-Equivalence as Conjunction of Disjunction with Negation of Conjunction

$\neg \left ({p \iff q}\right) \dashv \vdash \left({p \lor q} \right) \land \neg \left({p \land q}\right)$

That is, negation of the biconditional means the same thing as either-or but not both, that is, exclusive or.


Non-Equivalence as Conjunction of Disjunction with Disjunction of Negations

$\neg \left ({p \iff q}\right) \dashv \vdash \left({p \lor q} \right) \land \left({\neg p \lor \neg q}\right)$