# Non-Equivalence

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## Contents

## 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

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