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Edexcel C3 question

In the specification it says:

Preamble
Methods of proof, including proof by contradiction and disproof by
counter-example, are required. At least one question on the paper
will require the use of proof.


I've never seen a proof by contradiction/disproof by counter-example unless I have but didn't realise, can anyone give an example of when it's come up on a paper?
When it says at least one question will require proof

Is that e.g.
http://www.examsolutions.net/a-level-maths-papers/Edexcel/Core-Maths/Core-Maths-C3/2011-January/paper.php question 2(a) or 2(b)

http://www.examsolutions.net/a-level-maths-papers/Edexcel/Core-Maths/Core-Maths-C3/2010-January/paper.php question 2(a)?

If not which question is it?

Thank you :biggrin:
Original post by IgorYakov
In the specification it says:

Preamble
Methods of proof, including proof by contradiction and disproof by
counter-example, are required. At least one question on the paper
will require the use of proof.


I've never seen a proof by contradiction/disproof by counter-example unless I have but didn't realise, can anyone give an example of when it's come up on a paper?
When it says at least one question will require proof

Is that e.g.
http://www.examsolutions.net/a-level-maths-papers/Edexcel/Core-Maths/Core-Maths-C3/2011-January/paper.php question 2(a) or 2(b)

http://www.examsolutions.net/a-level-maths-papers/Edexcel/Core-Maths/Core-Maths-C3/2010-January/paper.php question 2(a)?

If not which question is it?

Thank you :biggrin:

Neither of those papers contains a proof-by-contradiction or a counterexample question, as far as I can tell.
The canonical example of a proof by contradiction is Euclid's proof that there are infinitely many primes, which is in the spoiler (but if you want to practise contradiction, you might like to leave this as an exercise, with the hint that "you can create a number which is not divided by any numbers in a given list, by multiplying the list together and adding 1").

Spoiler



I don't think I've ever seen it on a paper, that I remember (I took A-levels two years ago, so my memory may be faulty). It's quite hard to practise, too.
Original post by IgorYakov
In the specification it says:

Preamble
Methods of proof, including proof by contradiction and disproof by
counter-example, are required. At least one question on the paper
will require the use of proof.


I've never seen a proof by contradiction/disproof by counter-example unless I have but didn't realise, can anyone give an example of when it's come up on a paper?
When it says at least one question will require proof

Is that e.g.
http://www.examsolutions.net/a-level-maths-papers/Edexcel/Core-Maths/Core-Maths-C3/2011-January/paper.php question 2(a) or 2(b)

http://www.examsolutions.net/a-level-maths-papers/Edexcel/Core-Maths/Core-Maths-C3/2010-January/paper.php question 2(a)?

If not which question is it?

Thank you :biggrin:


A simple proof by contradiction is the proof that 2 \sqrt 2 is not a rational number.
Proof. Assume that it is i.e. 2=ab \sqrt 2=\frac{a}{b} where a and b are integers without a common factor.
Then squaring both sides 2=a2b2a2=2b2 2=\frac{a^2}{b^2} \Rightarrow a^2=2b^2 so a2a^2 is an even number a\Rightarrow a is an even number.
So put a=2k a=2k then we have 4k2=2b2b2=2k2b2 and hence b4k^2=2b^2 \Rightarrow b^2=2k^2 \Rightarrow b^2 \mathrm{\ and\ hence\ }b are even numbers.
So a and b are both even numbers which contradicts our assumption that they had no common factor. So our initial assumption is incorrect and 2 \sqrt 2 cannot be expressed as a fraction.
A simple disproof by counterexample is the following.
Prove that the statement x2>4x>2x^2>4 \Rightarrow x>2 is false.
Proof. x=3x2=9>4 but 3x=-3 \Rightarrow x^2=9>4 \mathrm{\ but\ }-3 is not >2>2
(edited 9 years ago)
Reply 3
Original post by Smaug123
Neither of those papers contains a proof-by-contradiction or a counterexample question, as far as I can tell.
The canonical example of a proof by contradiction is Euclid's proof that there are infinitely many primes, which is in the spoiler (but if you want to practise contradiction, you might like to leave this as an exercise, with the hint that "you can create a number which is not divided by any numbers in a given list, by multiplying the list together and adding 1").

Spoiler



I don't think I've ever seen it on a paper, that I remember (I took A-levels two years ago, so my memory may be faulty). It's quite hard to practise, too.



Original post by brianeverit
A simple proof by contradiction is the proof that 2 \sqrt 2 is not a rational number.
Proof. Assume that it is i.e. 2=ab \sqrt 2=\frac{a}{b} where a and b are integers without a common factor.
Then squaring both sides 2=a2b2a2=2b2 2=\frac{a^2}{b^2} \Rightarrow a^2=2b^2 so a2a^2 is an even number a\Rightarrow a is an even number.
So put a=2k a=2k then we have 4k2=2b2b2=2k2b2 and hence b4k^2=2b^2 \Rightarrow b^2=2k^2 \Rightarrow b^2 \mathrm{\ and\ hence\ }b are even numbers.
So a and b are both even numbers which contradicts our assumption that they had no common factor. So our initial assumption is incorrect and 2 \sqrt 2 cannot be expressed as a fraction.
A simple disproof by counterexample is the following.
Prove that the statement x2>4x>2x^2>4 \Rightarrow x>2 is false.
Proof. x=3x2=9>4 but 3x=-3 \Rightarrow x^2=9>4 \mathrm{\ but\ }-3 is not >2>2




The first one is something I've never seen before but it makes sense, I'm not sure how to practice these and I don't know why it says at least one question will require proofs, I've not seen any at all so I don't know what type of things they expect :/
Reply 4
Original post by IgorYakov
The first one is something I've never seen before but it makes sense, I'm not sure how to practice these and I don't know why it says at least one question will require proofs, I've not seen any at all so I don't know what type of things they expect :/


Out of all the C3 paper I've ever done I've never seen proof by contradiction, counterexample could be likely but as shown that is straightforward.

Proof isn't even in the C3 textbook so I wouldn't worry about it tbh,
Reply 5
Original post by forsparta
Out of all the C3 paper I've ever done I've never seen proof by contradiction, counterexample could be likely but as shown that is straightforward.

Proof isn't even in the C3 textbook so I wouldn't worry about it tbh,



Yeah true but it's in the spec and they might throw one in there this year to give some people a shock :tongue:
[QUOTE="IgorYakov;47810512"]The first one is something I've never seen before but it makes sense, I'm not sure how to practice these and I don't know why it says at least one question will require proofs, I've not seen any at all so I don't know what type of things they expect :/[/QUOTE

For proof by contradiction we make use of the fact that if you start from something that is true then it is impossible (by mathematically correct steps) to finish up with something that is false.
So if we are asked to prove that A=B, we start by assuming A does not equal B and show that it leads us to a contradicion.
The simplest example I cxan come up with is
Prove that x2+12xx^2+1 \geq 2x for all values of xx
So we start by assuming that x2+1<2xx2+2x+1<0x^2+1<2x \Rightarrow x^2+2x+1<0
But x2+2x+1=(x+1)20x^2+2x+1=(x+1)^2 \geq 0 contradicting our assumption. So our assumption must be incorreect and hence x2+12xx^2+1 \geq 2x
Reply 7
Original post by brianeverit


For proof by contradiction we make use of the fact that if you start from something that is true then it is impossible (by mathematically correct steps) to finish up with something that is false.
So if we are asked to prove that A=B, we start by assuming A does not equal B and show that it leads us to a contradicion.
The simplest example I cxan come up with is
Prove that x2+12xx^2+1 \geq 2x for all values of xx
So we start by assuming that x2+1<2xx2+2x+1<0x^2+1<2x \Rightarrow x^2+2x+1<0
But x2+2x+1=(x+1)20x^2+2x+1=(x+1)^2 \geq 0 contradicting our assumption. So our assumption must be incorreect and hence x2+12xx^2+1 \geq 2x


Aah right, thanks a lot :biggrin:
Original post by IgorYakov
Aah right, thanks a lot :biggrin:


No problem.

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