Question

Verify that the following are tautologies by citing the appropriate previous result. Do not make truth tables. My attempts will come as attachments shortly. you may view the questions in the mean time

Answer 1

@UnkleRhaukus how do I do a proof by equivalence?

Answer 2

The next two attachments are for 1.4.16

Answer 3

use logical equivalencies for example De Morrgan's law

Answer 4

Which one first ?

Answer 5

well how do I use De Morrgan's law for 1.4.15 ? and the 1.4.16 just needs to be checked that's about it.

Answer 6

maybe something like this? \[\begin{align*}\neg (\neg P)&\iff P\\
\neg\neg (\neg P)&\iff \neg P\\
(\neg P)&\iff \neg P\\\\&\text {True}
\end{align*}\]

What were the previous results the question talks about ?

Answer 7

I have no idea what that was to be honest... x,x

Answer 8

it sucks that I can't do truth tables on this one otherwise bam...done.

Answer 9

citing the appropriate previous result .............. WHAT IS THAT? D:

Answer 10

i was hoping you might have had that \(\neg \neg A=A\)

Answer 11

:S so strange... maybe we can move to 1.4.16 just to see if I've done this correctly...

Answer 12

OH! what if I modify parts of the problem from 1.4.15 to make it a tautology? I've read the sentence about if you put the negation on one side, it produces a contradiction

Answer 13

i just read that bit too, tight at the end of Ex1.4.15,

they should have put that instruction at the top of the exercise

Answer 14

so maybe if I can somehow produce a new .... a new tautology by ummm... hmmmmmm... adding negation signs in random parts of the problem??

Answer 15

i think that is what you have to do

Answer 16

so do I add negation signs all over the place? for the rest of the problems...

Answer 17

So maybe something more like this

For this to be true
\[\neg (\neg P)\iff P\]requires
\[\neg (\neg P)\iff \neg P\]to be false

\[P \iff \neg P\]which it is

Answer 18

i dont really think i have the format quite right,
do you have any examples from class?

Answer 19

hmmmmmm let me check

Answer 20

only implies.... I know that... equivalent... . that's easy too...truth table related

Answer 21

are you meant to use Ex1.4.14 to help you with Ex1.4.15 ?

Answer 22

I know that a tautology is everything is T in the table...contradiction is everything is false. If I can't use a truth table on this one, the only thing I could think of is modify the daylights out of it.
I wasn't assigned 1.4.14....I had to do 1.4.15 and 1.4.16

Answer 23

we could just skip 1.4.15 for now and just check on 1.4.16 since that problem allows truth tables.

Answer 24

i think Ex1.4.15 is kinda like proof by contradiction .

Answer 25

Lets do the truth tables ones then ,

Answer 26

o-0 how is that a proof by contradiction if I have to verify that there's a tautology and I can't draw a table cuz it ain't allowing me.