NOR is Commutative/Proof 1

Theorem

$p \downarrow q \dashv \vdash q \downarrow p$


Proof

By the tableau method of natural deduction:

$p \downarrow q \vdash q \downarrow p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \downarrow q$ Premise (None)
2 1 $\neg \paren {p \lor q}$ Sequent Introduction 1 Definition of Logical NOR
3 1 $\neg \paren {q \lor p}$ Sequent Introduction 2 Disjunction is Commutative
4 1 $q \uparrow p$ Sequent Introduction 3 Definition of Logical NOR

$\Box$


By the tableau method of natural deduction:

$q \downarrow p \vdash p \uparrow q$
Line Pool Formula Rule Depends upon Notes
1 1 $q \downarrow p$ Premise (None)
2 1 $\neg \paren {q \lor p}$ Sequent Introduction 1 Definition of Logical NOR
3 1 $\neg \paren {p \lor q}$ Sequent Introduction 2 Disjunction is Commutative
4 1 $p \downarrow q$ Sequent Introduction 3 Definition of Logical NOR

$\blacksquare$

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.