Contrapositive of a Conditional Statement

The contrapositive of "p implies q" is "not q implies not p". It looks quite different, but in fact is logically equivalent to the original conditional, which means we can use them interchangeably.

Learning Objectives:
1) Identify the logical structure of the contrapositive to a conditional
2) Rewrite a sentence in it's contrapositive form
3) Use that the contrapositive is logically equivalent to a conditional

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YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!

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This video was created by Dr. Trefor Bazett

[skellingtonmeteoryballoon]

After this class, I was seeing p's and Q's in everything. Lol

[vikranttyagiRN]

Though these videos are short but they are incredibly precise and up to the point. Great Stuff

[Hooghog]

Great explanation, thank you! The following statements seemed to stick with me a bit more in case they're helpful for anyone:
p = 'I study hard', q='I will pass'

1) p->q = 'If I study hard, then I will pass' (sucks if I study hard and then _don't_ pass, but I'm satisfied with the other cases)
2) ~q->~p = 'If I didn't pass, then I didn't study hard' (sucks if I didn't pass and I actually _did_ study hard, but I'm satisfied with the other cases)
3) q\/~p = 'Either I passed, or I didn't study hard' (sucks if I didn't pass and I actually _did_ study hard, but I'm satisfied with the other cases)
4) ~p\/q = 'Either I didn't study hard, or I passed' (sucks if I studied hard and then I _didn't_ pass, but I'm satisfied with the other cases)

Regardless that statement (1) was said before the exam and statements (2-4) were said after,
They are ALL routed in the belief of 'If I study hard enough, then I will pass'. They are therefore all logically equivalent

[cleverqwerty]

Have exams starting next week which have absolutely nothing to do with discrete math.... And here I am... hooked to this playlist...

[vaibhavshahi]

Your videos are damn good for clearing the concepts up to the point

[ShoaibAli-erid]

You just solved my problem, I was scratching my head whole day having problem in implications, contrapositive, double arrow...but all got easy for me...thank you sir

[MohamedElsayed-fvkp]

The videos are short, but the value is incredible. Thank you so much .

[alterwinters]

Short Video not 10-20 mins and fast to learn in 4 mins tysm!

[djtygre]

Really helpful about the grammar issues (also overlooked), also the either / or way of saying it.

[justinhansen]

Great job on the video, it helped out a bunch.

[MichaelFaradayStnine]

Keep it your response lecturer
, you very well done ✅

[bobtate]

Was it your intent to say that "~p v q" is the same as saying "EITHER ~p or q is true"? Doesn't the use of "either/or" imply that ~p and q can't both be true (xor)?

[humanrightsadvocate]

1. If n² is even then n is even.
2. Either n² is odd or n is even.
3. If n is odd then n² is odd.
4. Either n is even or n² is odd.
Proving either of these sentences proves them all. Proving statement 3 is trivial. (2k+1)² = 4k² + 4k + 1 = 2(2k²+2k) + 1. But we can consider 2k²+2k = q. And we know q is some integer because of the Closure Properties of Addition and Multiplication over the set of integers. So, we know statement 3 is true.

Let's check. Statement 1. n² = 2, then n is even. Wait! Where did I go wrong?

[minhthien]

Does ~p->~q equivalent to ~(p->q)? Using truth table I see what they are not equivalent, but I thought negation could be distributed?

[ironsfamily]

Great videos! I tend to have trouble efficiently learning from a textbook so these are a great help as I prepare for today's quiz.
Side not: Does his voice cadence remind anyone else of Veritasium (the beard helps too I'm sure, lol)

[tannudadhich]

He isn't only teaching Math, he's also teaching English grammer 😊😊😊😊 and Best thing about this tutorial is videos is short and on point.

I have better example:
If I watch this Tutorial, then I'll pass. P ⇒ Q
Either I didn't watch this tutorial, or I pass. ~ P ∨ Q
If I don't pass, then I didn't watch this Tutorial. ~Q ⇒ ~P
Either I pass or I didn't watch this tutorial. Q ∨ ~P

[noxx]

My only question is you said it's wrong to output. If I pass, then I studied hard as that implies passing is contingent on studying hard, but you put if I don't pass, then I didn't study hard that's the same thing is it not?

[tatyana]

hello, im having trouble understanding the contrapostive section. is the contrapositive (>q --> >p)?

[insertnamehere]

Why does xp stay xp when taking the contrapositive of xq implies xp?

[lanzibangli]

New meaning to "stay on your Ps and Qs"