PROOF by CONTRAPOSITION - DISCRETE MATHEMATICS

We look at an indirect proof technique, Proof by Contraposition.

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[syremusic_]

(I worked out the 7x+9 proof more thoroughly)


Prove: If 7x+9 is even, then x is odd.
Let's use contraposition.
If x is even, then 7x+9 is odd.
Assume x is even.
x=2k, where k is some integer.
7(2k)+9=14k+9
=14k+8+1 (breaking up the 9 into 8+1 to "properly" show that it's odd)
=2(7k+4)+1
7x+9 is odd!
By contraposition, we have proved our original statement.
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[Splashmasterson]

I will always remember your raining solution "Hey, there's some dude outside and he's dry, therefore it isn't raining". I get it now!!!

[II_xD_II]

3:35
damn those curves
best logic I've ever seen

[BestLeaks]

Why can't you just use X, Y, Z? You are making things harder to understand.

[rwkar]

x, y would have made that too easier, that symbol's quite difficult to understand

[bekkiiboo]

Hey Trev, I have an interesting question. Suppose x is an element of the real numbers. If x^3-x > 0, then x > -1. I tried turning this into a logic statement:

(x^3-x > 0) --> (x > -1)

And then take the contrapositive:

(x < -1 V x = -1) -> (x^3 - x < 0 V x^3 - x = 0)

My question is this: I understand that only one of antecedents must be true, but would we do four separate cases? Does only one of those four have to be correct? Can I combine the consequent into one statement and assume only one of the antecedents, and assuming more than one possibility for x arises, how would we deal with that?

ALSO:

Suppose x is an element of the real numbers. If x^5-4x^4+3x^3-x^2+3x-4 > 0 or = 0, then x > 0 or x = 0.

Contrapositive would be: (x < 0) --> (x^5-4x^4+3x^3-x^2+3x-4 < 0),

So what I did, I added all of the even powers to the other side, thus resulting in:

x^5+3x^3+3x < 4x^4+x^2+4

Then I said, "If x is negative, then for all x^n, with n an element of Z, it follows that for odd n's, x^n < 0, and for all even n's, x^n > or = 0, therefore the contrapositive is true. Does that sound like a thorough proof?

[edemcudjoe]

I think we'd all appreciate it more if you could please use x or Y or Z Kai and Physi is a bit confusing

[FengkieJunis-]

Where and when we should use a certain proofing method over another???

[kuihaochang]

How to use "suppose"? Is it as same as "assume"?

[jasonwong]

wait so why do we need to do proof by case at 4:51. ? shouldn't you only need to prove one or the other? since it's an OR

[psn]

if you are going to use greek symbols please make sure you write NEATLY

[FabianAmran]

What is this si and fi you speak of. I feel that you are cluttering my mind with non essential mnemonics

[Prince-olzk]

When you you prove by cases, do you show that both cases result in the same truth value for the proof to be right or can one case work?

[FluxProGaming]

Why is it that we have to do a proof by case for the OR statement with the conditional. Wouldnt it be that if one of them is true, the brackets with OR is also true. I would understand for an AND operator to do a proof by case to prove that both are true which would make the bracket statement true as well. A little confused about this....hopefully you can answer.

[scent_from_another_realm]

Hey Trev, I was given the task: Write the following arguments symbolically and determine which of them are valid:
All birds can fly.
Pigs cannot fly.
Therefore, pigs are not birds.

So I said
P(x): x is a bird
Q(x): x can fly

Vx(P(x)->Q(x))

let y be a pig

P(y) does not imply Q(y)

therefore

P(y) is not equal to P(x)

therefore

pigs are not birds.

Therefore arguement is valid

Is this completely okay as an answer?

[EquinoXReZ]

Why does the or statement make it become proof by case? Wouldn't we only need to prove one of them to show that its true?

[shaneshort]

I appreciate these videos, and I understand these videos are old and this won't even matter at this point, but I don't understand why you didn't just use simpler variables like x and y that we use daily in English instead of using the Greek alphabet when it is completely unnecessary, and to top it off you are using Phi and Psi which in this Crayola crayon drawing format look very similar. Like, why add the extra complication to something people are barely grasping as it is? I get they have their place in math, but like, c'mon... It just feels a bit overboard at this point rather than making a conducive learning environment.

[rusiruchapana]

Your handwriting s are very difficult to finding

[AskKicker]

I like the laziness, because fuck discrete, it's intuitive as fuck without all the stupid letters and shit.

[stevedl]

It seems to me, then, that if this or that then the other.
If not the other, then not this or that.
Not the other, then not this and not that.
I`m not knocking it - I`ll have it come year`s end.
Oh, the bob`s just dropped - `or` being inclusive.