# Propositional logic question

1. Oct 21, 2013

### phospho

Negate $[\neg (p\wedge \neg q)]\wedge \neg r$

and relpace the resulting formula by an equivalent which does not involve $\neg, \vee, \wedge$

attempt:

$\neg ([\neg (p\wedge \neg q)]\wedge \neg r) = \neg \neg (p \wedge \neg q) \vee \neg \neg r$

$= (p \wedge \neg q) \vee r$
$= (p \vee r) \wedge (\neg q \vee r) = \neg ((p\vee r) \implies \neg (\neg q \vee r))$
$= \neg ((p \vee r) \implies \neg \neg q \wedge \neg r) = \neg ((p \vee r) \implies q \wedge \neg r)$

$= \neg (\neg p \implies q \implies \neg (\neg q \implies r))$

not sure if this is correct so far, and if it is, where to go from here

2. Oct 21, 2013

### FeDeX_LaTeX

You are very close at this point; try replacing $p \wedge \neg q$ with an equivalent statement not involving $\wedge$.

3. Oct 21, 2013

### phospho

I'm sorry I can't find one :( I'm having a lot of troubles figuring out how to find equivalent statements. I have some written down from lectures, but none which I can think of help here.

4. Oct 21, 2013

### phospho

all I have is $p \wedge \neg q = \neg (\neg p \wedge q )$ but I don't know if you can "factor out" $\neg$

5. Oct 21, 2013

### FeDeX_LaTeX

Hint:
$a \wedge b \equiv \neg(a \implies \neg b)$

6. Oct 21, 2013

### phospho

so $(p \wedge \neg q ) \vee q = \neg (p \implies \neg \neg q) \vee r = \neg \neg (p \implies q) \implies r) = (p \implies q) \implies r$ ?

I am very curious on how you know all of these equivalent statements? Also, are you allowed to multiply out as in algebra? e.g. does $\neg (a \vee \neg b ) = \neg a \vee \neg \neg b$ ?

7. Oct 21, 2013

### FeDeX_LaTeX

You could check your answer with a truth table and see if that matches up with what you'd expect.

In general, no, the distributive law does not necessarily work in the way that you are familiar with. For instance, $\neg(p \vee q) \neq (\neg p) \vee (\neg q)$.

8. Oct 21, 2013

### phospho

hm I see,

the thing is, I see online a lot of people just stating equivalent statements, but I'm not sure how they just "know" it. I do draw truth tables, but I don't want to be doing that all the time. Is there something I'm missing?

Thank you for your help btw, is my answer correct?

9. Oct 21, 2013

### FeDeX_LaTeX

It helps to know the negation rules (De Morgan's laws) and some identities for replacing conjunctions, disjunctions, implications, and so forth. In calculus you are likely familiar with several identities or formulae which you have learned and have become second nature to you through practice; the same can be true for propositional and predicate logic.

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