Hi Mr Bicen, When it comes to proofs and proof questions(AS and A2) in general, I find them quite tricky in the sense that they can literally ask you to prove anything and there doesn't seem to be a concrete method to solving them. I understand how they want us to structure our answers for example the start by assuming initial statement is false etc. etc. but the bits in between are quite tricky sometimes. Are these questions usually low tariff? Or have you seen them thrown in for a lot of marks in your experience? Just asking to prioritize my revision as I don't want a potentially small topic bottleneck my preparation for the exams. Thanks again for all your videos, super helpful!
Great question! These are usually 4 marks top for a proof by contradiction - and you can definitely secure the first mark but writing the opposite statement as your assumption! You can then move on and make sure you get other marks elsewhere, as sometimes that’s an excellent exam strategy, especially when the grade boundaries are not exactly high… it’s a bit like trying to game the exam! If you want to get good, you can just practice and practice and you’ll start spotting more patterns that at first seem like you just need to know what to do - it’s actually a few main options. Hope that helps! Good question!
I this correct 1) Statement 2) Assume the statement is false 3) Prove contradiction of statement is false 4) Assumption is wrong, therefore the statement is true
Also the assumption around 6:00 should be that there exists an integer n such that n squared is even and n is odd. Otherwise you could just use a counterexample to show a contradiction.
Hi Mr Bicen for proving that there is no integers a and b for which 25a+15b=1 can we do proof by exhaustion after making the assumption that there IS. for example i considered a and b to be two different even number, two different odd numbers, one odd and one even - would this be wrong
Hi there, it is because you have joined at the incorrect tier - members videos are only available to those who sign up at Further Maths Options or above! It’s cheaper to sign up directly on TH-cam rather than on iOS by the way! Should be £23.99
In order is simplest, but if you wanted to tackle the more challenging things first, I’d recommend doing trig, then differentiation, then integration - you then have longer to get your head around some of these concepts, and you leave some of the simpler ones until the end. You’ll need to do Ch 1 and 2 though before the ones I suggest, as they come up later on.
I'm not sure what you mean - there isn't a need to prove that n+2 is undefined, as it is defined! It's 2 more than n, and so must also be odd, which is the contradiction, as it is bigger than n.
I think this topic is really interesting, and so different to the rest of A-Level maths! It's similar to a lot of the idea/approaches in a maths degree!
"Prove that there is no greatest odd integer". Your assumption negated and proved that there exist a greater odd integer. How it is concluded as wrong? What am I missing here?
Because the thing I assumed (that there WAS a greatest odd integer) was shown to be nonsense, meaning that the thing I assumed was incorrect. Hence there is no greatest odd integer!
By far the hardest topic in a level maths
I agree. Its the only topic that doesn't have the crank-the-handle style of question you'd typically see on other pure topics.
Hi Mr Bicen, When it comes to proofs and proof questions(AS and A2) in general, I find them quite tricky in the sense that they can literally ask you to prove anything and there doesn't seem to be a concrete method to solving them. I understand how they want us to structure our answers for example the start by assuming initial statement is false etc. etc. but the bits in between are quite tricky sometimes. Are these questions usually low tariff? Or have you seen them thrown in for a lot of marks in your experience? Just asking to prioritize my revision as I don't want a potentially small topic bottleneck my preparation for the exams. Thanks again for all your videos, super helpful!
Great question! These are usually 4 marks top for a proof by contradiction - and you can definitely secure the first mark but writing the opposite statement as your assumption! You can then move on and make sure you get other marks elsewhere, as sometimes that’s an excellent exam strategy, especially when the grade boundaries are not exactly high… it’s a bit like trying to game the exam! If you want to get good, you can just practice and practice and you’ll start spotting more patterns that at first seem like you just need to know what to do - it’s actually a few main options. Hope that helps! Good question!
I this correct
1) Statement
2) Assume the statement is false
3) Prove contradiction of statement is false
4) Assumption is wrong, therefore the statement is true
Yep! In 2), the opposite of the statement is called the negation. Well summarised! 👏🏼
Proof 1 • Proof by Contradiction done! brilliant video!
You are a very smart teacher❤
Thank you for the simplest explanation on this
Slight correction, the sqaure root of an even number can only be assumed to even if it is already an integer.
Also the assumption around 6:00 should be that there exists an integer n such that n squared is even and n is odd. Otherwise you could just use a counterexample to show a contradiction.
Hi Mr Bicen for proving that there is no integers a and b for which 25a+15b=1 can we do proof by exhaustion after making the assumption that there IS. for example i considered a and b to be two different even number, two different odd numbers, one odd and one even - would this be wrong
I don’t think this would be accepted. Remember that an and b could also be negative.
Hi Mr Bicens I am already a Channel Member. However I am unable to access Members Videos.
Hi there, it is because you have joined at the incorrect tier - members videos are only available to those who sign up at Further Maths Options or above! It’s cheaper to sign up directly on TH-cam rather than on iOS by the way! Should be £23.99
Sir what’s the best route to take for the chapters for year 2 pure
I’d just do it in order
In order is simplest, but if you wanted to tackle the more challenging things first, I’d recommend doing trig, then differentiation, then integration - you then have longer to get your head around some of these concepts, and you leave some of the simpler ones until the end. You’ll need to do Ch 1 and 2 though before the ones I suggest, as they come up later on.
4:16 but you havent proved that n+2 is not undefined, so wouldnt the proof be not rigorous enough?
I'm not sure what you mean - there isn't a need to prove that n+2 is undefined, as it is defined! It's 2 more than n, and so must also be odd, which is the contradiction, as it is bigger than n.
at 10:20 could n = 2k, so when you square n you get an even number, which would disprove the statement as n is an even number.
It asked for contradiction, however, so we do need to prove it using n as an odd number!
Such an interesting topic, considering this is just out of the loop of what the other maths are
I think this topic is really interesting, and so different to the rest of A-Level maths! It's similar to a lot of the idea/approaches in a maths degree!
Is this the only on year 2 proofs?
Yep contradiction is Year 2 only.
at 8:45 i dont understand where 2k+1 comes from
We make that up ourselves. If k is an integer (which we should technically state) then 2k+1 is an odd number!
for 10:25 could we say 2 k squared when the integer is 2k is not odd. So, n must be even?
I’m not quite sure what you mean by this - can you rephrase?
"Prove that there is no greatest odd integer". Your assumption negated and proved that there exist a greater odd integer. How it is concluded as wrong? What am I missing here?
Because the thing I assumed (that there WAS a greatest odd integer) was shown to be nonsense, meaning that the thing I assumed was incorrect. Hence there is no greatest odd integer!
blCK AND SMOOTH AND CRUNCHY
easier then Proof by Induction, oh i hate that topic.