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?
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
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.
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.
Im not sure if this is a dumb question but I was working on the last example problem and was wondering if my way of proving is correct. When you had the equation 3n+2 = 6k+5, could you subtract 2 from the left side to get 3n= 6k+3, divide the whole equation by 3 to get n=2k+1 to prove your proof? I'm not sure if I'm just thinking too algebraically or if my method of proving is a correct approach, so I'd appreciate help from anyone with more knowledge.
Ok it’s been 3 weeks, but that’s too algebraic, normally stick to one side of the equation since we’re not trying to balance the equation so to speak. Instead transforming the first equation into the second through substitution
@@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
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.
@@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
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?
Very thankful for your videos! You give great examples and clearly show you steps. We all owe you one!
Thank you so much. I am going to watch your videos this whole semester.
These videos are a life saver
Thank you so much for these series. Such a clear articulation of these concepts; so approachable. Cheers
I dont understand the math 5:25, how're you allowed to arrange it like that?
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?
The definition of an odd integer is that it can be written in the form 2n+1 where n is an integer. That is why we must do the rewrite.
What about in the form of 2n+(some other odd integer other than 1 let's say 3) would that still be valid or it only has to be in the form of 2n + 1. 😅
Best and simple explanations 🔥🔥🔥🔥
It's straightforward with the most common examples.
I love this lady she is amazing.
Thanks!
In the first problem, if p was negated, wouldn’t that make n not an integer? Which conflicts with q because even numbers are integers.
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
Helpful video and thank you so much for explaining in simple terms ❤kee p going on
THIS IS SO HELPFUL I LOVE YOU
love this playlist!
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.
@@SawFinMath brilliant, thank you professor!
YOU'RE SO HELPFUL! THANK YOUUUUUU
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.
Im not sure if this is a dumb question but I was working on the last example problem and was wondering if my way of proving is correct. When you had the equation 3n+2 = 6k+5, could you subtract 2 from the left side to get 3n= 6k+3, divide the whole equation by 3 to get n=2k+1 to prove your proof? I'm not sure if I'm just thinking too algebraically or if my method of proving is a correct approach, so I'd appreciate help from anyone with more knowledge.
Ok it’s been 3 weeks, but that’s too algebraic, normally stick to one side of the equation since we’re not trying to balance the equation so to speak. Instead transforming the first equation into the second through substitution
@@zachhogan255 Thank you!
Hi Kim. Thank you for all the videos. I had a question. How did 2( 3k +2) +1 get to 2r + 1, r = 3k +2 ... ? Where did the 'r' come from?
If you don't know by now, the 'r' is just used to represent "(3k + 2)" so it's easier to see the form it's in.
We assume r=3k+2 to conclude
@@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
Thank you. Your the reason i got a good grade in my class
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.
Think about it as 6k+4+1. The 6k+4 factors to 2(3k+2) making it an even integer. Then add the 1 we left off, which makes it odd.
@@SawFinMath MATH MAGIC🥲
thanks, really helpful !
how can you know that this method is applicable to your question
This makes 10x more sense than my text book (I dont get any kind of lecture in this class) and I still feel like I am trying to learn greek...
Proof is tough. Just keep practicing and it will get easier
@@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
thank you
You're welcome
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?
Don't be too hard on yourself
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