Author Topic: Anyone have any idea how to solve this using 'laws of logic'? (Read 1023 times) Tweet Share

saifh · « **on:** March 25, 2014, 08:58:55 pm »

As the title says, can anyone solve this following question using the laws of logic?

The link with the question is attached to this thread, as I have no idea how to use special characters on this forum.

Any help is welcomed, as I'm stuck!

Thanks.

brenden · « **Reply #1 on:** March 25, 2014, 09:05:03 pm »

Does 'truth-table' or 'tableaux' mean anything to you?

saifh · « **Reply #2 on:** March 25, 2014, 09:07:13 pm »

Quote from: Arnold Brendenegger on March 25, 2014, 09:05:03 pm

Does 'truth-table' or 'tableaux' mean anything to you?

I know what a truth table is but you can't use it, you have to use these certain rules.

And no idea what a tableaux is.

brenden · « **Reply #3 on:** March 25, 2014, 09:12:32 pm »

Tableaux is an infinitely more efficient method of determining in what case a statement/argument is true/valid than using a truth-table. I can only assume that 'laws of logic' isn't referring to tableaux if you don't know what one is. If it's neither of those then I'm not sure what laws of logic is referring to. I'm sure someone else will, though. What subject is this for (are you still in high school?)?

Edit: http://en.wikipedia.org/wiki/Propositional_logic#Basic_and_derived_argument_forms

Skimming that might help to see if it clicks in your mind/reminds you about something your teacher mentioned.

b^3 · « **Reply #4 on:** March 25, 2014, 10:09:51 pm »

These are the three laws you'll need to play around with, see if you can get it before you look in the spoiler.
$\begin{alignedat}{1}\text{Implication Law} \\ a\rightarrow b & \equiv a\vee\neg b \\ \text{De Morgan's Laws} \\ \neg\left(a\wedge b\right) & \equiv\left(\neg a\right)\vee\left(\neg b\right) \\ \text{Distributive Law} \\ a\wedge\left(b\vee c\right) & \equiv\left(a\wedge b\right)\vee\left(a\wedge c\right) \end{alignedat}$

Spoiler

$\begin{alignedat}{1}\text{By the Implication Law} \\ \left(\neg p\vee\neg q\right)\implies\neg q & \equiv\neg\left(\neg p\vee\neg q\right)\vee\left(\neg q\right) \\ \text{By De Morgan's Laws} \\ & \equiv\left(\left(\neg\neg p\right)\vee\left(\neg\neg q\right)\right)\vee\left(\neg q\right) \\ & \equiv\left(p\wedge q\right)\vee\left(\neg q\right) \\ \text{By the Distributive Law} \\ & \equiv\left(\neg q\right)\vee\left(p\wedge q\right) \\ & \equiv\left(\neg q\vee p\right)\wedge\left(\neg q\vee q\right) \\ \neg q\vee q\equiv T & \:\:\text{\text{"not q or q"}} \\ & \equiv\left(\neg q\vee p\right)\wedge T \\ & \equiv\left(p\vee\neg q\right) \end{alignedat}$

saifh · « **Reply #5 on:** March 25, 2014, 11:05:21 pm »

Quote from: b^3 on March 25, 2014, 10:09:51 pm

These are the three laws you'll need to play around with, see if you can get it before you look in the spoiler.
$\begin{alignedat}{1}\text{Implication Law} \\ a\rightarrow b & \equiv a\vee\neg b \\ \text{De Morgan's Laws} \\ \neg\left(a\wedge b\right) & \equiv\left(\neg a\right)\vee\left(\neg b\right) \\ \text{Distributive Law} \\ a\wedge\left(b\vee c\right) & \equiv\left(a\wedge b\right)\vee\left(a\wedge c\right) \end{alignedat}$

Spoiler
$\begin{alignedat}{1}\text{By the Implication Law} \\ \left(\neg p\vee\neg q\right)\implies\neg q & \equiv\neg\left(\neg p\vee\neg q\right)\vee\left(\neg q\right) \\ \text{By De Morgan's Laws} \\ & \equiv\left(\left(\neg\neg p\right)\vee\left(\neg\neg q\right)\right)\vee\left(\neg q\right) \\ & \equiv\left(p\wedge q\right)\vee\left(\neg q\right) \\ \text{By the Distributive Law} \\ & \equiv\left(\neg q\right)\vee\left(p\wedge q\right) \\ & \equiv\left(\neg q\vee p\right)\wedge\left(\neg q\vee q\right) \\ \neg q\vee q\equiv T & \:\:\text{\text{"not q or q"}} \\ & \equiv\left(\neg q\vee p\right)\wedge T \\ & \equiv\left(p\vee\neg q\right) \end{alignedat}$

Didn't even look at the Distributive Law.

Got it without the spoiler, thanks!

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Author Topic: Anyone have any idea how to solve this using 'laws of logic'? (Read 1023 times) Tweet Share

saifh

Anyone have any idea how to solve this using 'laws of logic'?

brenden

Re: Anyone have any idea how to solve this using 'laws of logic'?

saifh

Re: Anyone have any idea how to solve this using 'laws of logic'?

brenden

Re: Anyone have any idea how to solve this using 'laws of logic'?

b^3

Re: Anyone have any idea how to solve this using 'laws of logic'?

saifh

Re: Anyone have any idea how to solve this using 'laws of logic'?

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