# 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

$\vdash \paren {\neg \paren {p \iff q} } \iff \paren {\paren {\neg p \land q} \lor \paren {p \land \neg q} }$

### 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

$\neg \paren {p \iff q} \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$