# 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

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

### Non-Equivalence as Conjunction of Disjunction with Negation of Conjunction

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

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

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