Logic 101 (#21): Applying Replacement Rules

Previously, we have seen how to use replacement rules in isolation. This lecture uses a succession of replacement rules to turn a very ugly statement into a very clean and digestible one.

In the process, we get a flavor of proofs, which will be the main topic two units from now.

[PunmasterSTP]

Replacement rules? More like “You are our schools.”, because you’ve helped so many of us out across so many institutions. Thank you so much!

[jaydearnoult]

thanks so much!! this is much easier to understand than the way my professor tried to explain it.

[MillerHighLifez]

This is an excellent series. Thank you so much for making this!

[abdiel]

1: ~(~Q => ~P)
2: ~(P => Q) -- contrapositive
3: ~(~(P ^ ~Q))
From here, I see two possible routes
A - 4: P ^ ~Q -- cancel-out the two negations in front; QED
-- OR --
B - 4: ~(~P v Q) -- DeMorgan's rule applied inside the first set of parentheses
B - 5: P ^ ~Q -- DeMorgan's rule applied to line B - 4; QED

Not sure if route A is correct, though. I'd go with route B since it seems to match the number of steps in the video at the 2:00 mark. I'll add onto this comment as I watch the video.


Note: I forgot to say how I got to step 3. It's what I thought was logically equivalent to the if-then statement.


After seeing the whole video, I see I misunderstood material implication (the thing that was "logically equivalent to the if-then statement"), though only slightly. Using DeMorgan's rule on ~(P ^ ~Q) that I used on line 3, I get back ~P v Q, which is exactly what you have for material implication on your line 3. I guess I didn't really mess up. I just thought of it differently.

Thanks for these videos, and keep it up!

[dosto_viski]

Sir i dont get the 4th one, if we apply demorgan to (~pvq) wouldn't that make ~[~(~p) ^ ~(q)], so therefore ~p^q ?

Thank you for classes.

[lakhaloth]

Wow this is ooo helpful!! saved my exam of logics in 2 weeks!

[eShredz]

Is this correct even if P and Q are flipped?
~(~Q=>~P)
~(~~Qv~P)(Implication Identity)
~(Q v ~P)(double negation)
(~Q ^ P)(DeMorgans)

[accountbertolini]

Couldn't you also say DeMorgan's work for <=> turning it into "Exclusive OR" ?

[hz]

Thank you so much brother thank you so much !!

[lea]

I don't understand why the negation of "or" is "and" and viceversa

[mohemodirar]

-(-q => -p) apply contraposition
-(p => q) apply material implication
-(-p or q) apply demorgans
p and -q

[agentmikster]

~(~Q=>~P): Original
~(P=>Q): Contrapositive
~(~P v Q): Since P=> Q is equal to ~P v Q
P ^ ~Q: Negate everything inside parentheses

[quaidnichols]

I think that you use a different form of logic than me. Contraposition=Transposition

[i.am.pier_]

why are the truth table not the same I'm losing it, everything is good but when it comes to the truth table they aren't the same, so in order for them to become equals we need to negate P instead of Q

[ZachBugay]

Would love to see you write this out or something, as opposed to just saying it. It would be way more helpful. Like buy a bamboo and use paint or something and write this out.

[kashabash]

Hmm..For some reason you sound different in this video.

[jvgama]

1. ~(~Q => ~P)
2. ~(P => Q)
3. ~(~P v Q)
4. ~~P ^ ~Q
5. P ^ ~Q

[mrsyed]

Hello William, I find your Logic 101 videos very helpful! I am trying to answer a question and I was hoping you or anyone could assist me through it.

The question requires me to simplify the following formula using the logical equivalencies.

¬ ((¬Q & ¬P) v ¬P)

[gidrengidrenovi]

~(~Q => ~P) MaIm
~(Q v ~P) DeMorg
~Q^P

[spoonstraw]

~(~Q => ~p)
<=>
~(Q v ~p)
<=>
~Q ^ p