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Conditional-disjunction equivalence:

(p $\implies$ q) $\equiv$ ($\lnot$p $\lor$ q)

To show:

($\lnot$q $\implies$ p) $\implies$ (p $\implies$ $\lnot$q) $\equiv$ ($\lnot$p $\lor$ $\lnot$q)

Attempt:

$\lnot$($\lnot$q $\implies$ p) $\lor$ (p $\implies$ $\lnot$q) by conditional-disjuction equivalence

$\equiv$ $\lnot$($\lnot$($\lnot$q) $\lor$ p) $\lor$ ($\lnot$p $\lor$ $\lnot$q) by conditional-disjunction equivalence

$\equiv$ $\lnot$(q $\lor$ p) $\lor$ ($\lnot$p $\lor$ $\lnot$q) by double negation law

$\equiv$ ($\lnot$q $\land$ $\lnot$p) $\lor$ ($\lnot$p $\lor$ $\lnot$q) by DeMorgan law

I am stuck at the last step.

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You are nearly there.

$(\neg q\wedge\neg p)\vee(\neg p\vee\neg q)\equiv[(\neg q\wedge\neg p)\vee\neg p]\vee\neg q$ by the associativity of $\vee$.

$\color{red}{[(\neg q\wedge\neg p)\vee\neg p]}\vee\neg q\equiv\neg p\vee \neg q$ by absorption on the red term.

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  • $\begingroup$ I Never saw ($\lnot$q $\lor$ $\lnot$p) as a single term to which the a law can be applied. Thank you. $\endgroup$ – Haslo Vardos Nov 7 '20 at 11:10
  • $\begingroup$ If you are satisfied with my answer consider accepting it by clicking the tick-mark button next to it. $\endgroup$ – Shubham Johri Nov 7 '20 at 11:11

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