i think the question can be reinterpreted as if 3n+2 is odd, then prove n is odd, all the while when n is an integer. essentially n being an integer is not really part of the proposition p (or q) and it's just a given
does the meaning of proving ~q>~p is true equals to proving that ~q implies ~p is a tautology and by the use of equivalent statement of p>q and ~q>~p , p>q is a tautology ,too? im so confused with this section and the previous.
@@joudialmarri no. We say that r = 3k + 2 because we are looking for the form of the definition for an odd number to prove the second example true through the use of contraposition. Here is the whole proof written out for the second example in this video. Prove “If n is an integer and 3n + 2 is even, then n is even Statement p: n is an integer and 3n + 2 is even Statement q: n is even Statement ¬q: n is odd Statement ¬p: n is an integer and 3n + 2 is odd Assume n is odd is true. By definition n = 2k + 1, k ∈ ℤ 3n + 2 = 3(2k + 1) + 2 3n + 2 = 6k + 3 + 2 3n + 2 = 6k + 5 3n + 2 = 6k + 1 + 4 3n + 2 = 6k + 4 + 1 3n + 2 = 2(3k + 2) + 1 3n + 2 = 2r + 1 where r = 3k + 2, r ∈ ℤ ∴ 3n + 2 is odd Since ¬q → ¬p is true, then p → q is true by contraposition. QED
@@SawFinMath i am spending all of my days off on this class trying to get it, I shared this playlist with the rest of my class to hopefully help others
Hi. I understand the method in the video, but how would you decide what prood to use? If the examples were listed in the textbook homework section, would it matter what method of proof I used - so long as I proved it?
Nope. You can prove many theorems in many ways. There is no right way, as long as the method you choose is complete. Typically textbooks will have you choose a specific way for practice in the section you learn that method in, but later on it is just whatever works. Find one you are super comfortable with and another you can use when your preferred method isn't panning out.
We have 6k+3+2, which is 6k+5, but I want to show it is an odd integer, so I wrote it as 6k+4+1 and factored a 2 from the 6k and the 4, making it 2(3k+2)+1
@@SawFinMath If I went from 6k + 3 + 2 to 6k + 5 as my final answer and say its odd. That would still be correct as that is basically in the form of n = 2k + 1 right?
Hello, I have a question. Do I need to learn Axioms before doing proof exercises? while doing the exercise, I realized that I needed to be aware of some axioms to do the proof. Those Axioms can be found on page 926 in the textbook. I'm pretty sure that had I paid attention in high school, I would be quite familiar with those axioms, but since I wasn't always a good student in highschool. So do you think that I need to read and learn those before I start doing proof exercises? How essential is it?
Alright. I was with you the whole way. Then at 5:35. you decide to pull some math magic. 6k + 3 + 2 -> 2(3k + 2) +1 . Nah. How the heck do you put the 1 on the outside there. I'm gonna need some laws or something to back that up. Because How can you just stick the remainder on the outside.
