A-Level Maths: A1-16 Proof by Contradiction Examples

[Kevin_Guo]

Thank you so much - this video is absolutely amazing and very clearly explained!

[floralsparkle]

for the second example could you just substitute n=2p+1 and then factorise to show that (4p^3 + 6p^2+3) is a multiple of 2 but the expansion also has a '+7' on the end and the sum of an odd and even number is odd so n is odd or would you need both the substitutions that you used (i hope this makes sense!)

[gradfathers]

GREAT!! I found it very simple to understand, keep it up!

[FATFILMSABLE]

Hi sir, I love these videos they are life savers! I was wondering if this playlist is going to cover mechanics and statistics? Thanks

[chapers]

For question two (2), can you do?:

n^3 + 5 = 2m + 1
n^3 = 2m - 4
n^3 = 2( m - 2 )
therefore n^3 is even
therefore n is even
contradiction as n cannot be even when n^3 + 5 is odd
if n^3 + 5 is odd, n must be even


thanks in advance if you can get round to this during your busy schedule of saving student's maths careers.

[chga]

Is it acceptable to say that if we assume true, 5a=1-15b and thus a=1/5 - 3b. Hence, if b is an integer then a cannot be an integer, hence a and b cannot be integers. Thanks

[allenokungbowa]

Sir, must we equate it to 2m +1
If we solve just the lefthand side and it is even doesnt that contradict the.assumed statement anyways.

[redroses]

How would we prove by contradiction, for integers a and b, where is bigger and equal 2 and a and b belong to real numbers that, either a does not divide b, or a does not divide b+1 ?

[albibarry]

Can you just make N=2m+1 substitute it into n^3+5 and show that the answer is even therefore when n is odd n^3+5 must be even

[skyeforster]

Hello, I was just wondering would the question specifically say prove by contradiction the following statement or would you have to assume to prove by contradiction because I feel like I would start proving by disproof or another type of proof.

[partnermammoth]

how do you differentiate which type of proof you should use?

[barksstuff]

Hello sir what do you think of my method:

For all intergers of n, if n^3 is odd n is not even

n^3 + 5 = 2a + 1
n3 = 2a -4
n3 must be even
thus n must be even

[jace]

Do you have any videos up on how to solve questions where the you have to prove rational - irrational = irrational, and proving there are no positive integer solutions to y^2 + x^2 = 1?

[abigailo]

Hi sir, for question 2, if you assumed both n^3 + 5 and n was even would it work the same cause its still assuming that the two statements aren't opposite.

[ibguxvi]

Would you still get marks if you factored out 4 (for the last question) so you were left with
4(2p^3+3p^2+2p)+2

and you mentioned that a number with a factor of 4 is an even because it also has a factor of 2, and an even number + 2 is still an even

[avi_mukesh]

For the first one, I rearranged the equation to get b=(1-5a)/15. The denominator as a product of its prime factors is 3*5 and because there is a 3, b must be irrational. Therefore, a must be irrational, and not an integer. Is this correct?

[_GShock_]

wait hold on, on number 2 it said n^3 + 5 is odd and n is even
if you assume both n^3 + 5 is odd and n is odd
im completely confused because if you wanted us to assume the opposite shouldn't the assumption be n^3 + 5 is even?

[freerights]

I saw a question that goes "There are no integer solutions to the equation x² - y² = 2"

So I did:
Let's assume that there are integer solutions to the equation x² - y² = 2, so:
x² - y² = 2
x - y = √2
If both x and y are integers, the subtraction of y from x would result in another integer, which contradicts the fact that x - y is √2, which is not an integer. Thus our assumption is incorrect, and the statement is true

Would this be sufficient to gain full marks for that question or would I have to do more?

In my textbook the answer was very long-winded, and factorised x² - y² - 2 into all of the possible combinations to see x and y would be integers in any of them, e.g:
[x - y = 2 and x + y = 1] or
[x - y = 1 and x + y = 2] and so on...

[chhayalad]

Why do you assume the second question is odd?

[baburlu]

But what if n^3+5 is even and n is even?