Non-Equivalence as Disjunction of Negated Implications

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Theorem

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


Proof 1

By the tableau method of natural deduction:

$\neg \left ({p \iff q}\right) \vdash \neg \left({p \implies q}\right) \lor \neg \left({q \implies p}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \left ({p \iff q}\right)$ Premise (None)
2 1 $\neg \left({\left ({p \implies q}\right) \land \left ({q \implies p}\right)}\right)$ Sequent Introduction 1 Rule of Material Equivalence
3 1 $\neg \left({p \implies q}\right) \lor \neg \left({q \implies p}\right)$ Sequent Introduction 2 De Morgan's Laws: Disjunction of Negations

$\Box$


By the tableau method of natural deduction:

$\neg \left({p \implies q}\right) \lor \neg \left({q \implies p}\right) \vdash \neg \left ({p \iff q}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \left({p \implies q}\right) \lor \neg \left({q \implies p}\right)$ Premise (None)
2 1 $\neg \left({\left ({p \implies q}\right) \land \left ({q \implies p}\right)}\right)$ Sequent Introduction 1 De Morgan's Laws: Disjunction of Negations
3 1 $\neg \left ({p \iff q}\right)$ Sequent Introduction 2 Rule of Material Equivalence

$\blacksquare$


Proof 2

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.


$\begin{array}{|cccc||ccccccccc|} \hline \neg & (p & \iff & q) & \neg & (p & \implies & q) & \lor & \neg & (q & \implies & p) \\ \hline F & F & T & F & F & F & T & F & F & F & F & T & F \\ T & F & F & T & F & F & T & T & T & T & T & F & F \\ T & T & F & F & T & T & F & F & T & F & F & T & T \\ F & T & T & T & F & T & T & T & F & F & T & T & T \\ \hline \end{array}$

$\blacksquare$


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