PROOF

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Proof by contradiction

To prove a statement by contradiction you first assume the statement in the question is false, this will be your assumption. Once you have stated your assumption you will attempt to prove your assumption correct. If don't correctly you should be unable to prove you assumption correct, in which you can then form a conclusion.

Example 1:

Prove by contradiction that there is no greatest odd integer. (We first start by assuming the statement is false)

Assumption: There is a greatest odd integer, n.

n + 2 > n (If we are suggesting that n is the great then n+2 is cannot be greater than n)

n + 2 = odd + even = odd (This contradicts the assumption as n+2 is odd and greater than n)

So there is a a greater odd integer than n, this contradicts the assumption of n being the greatest odd integer. There fore we can conclude there is no greatest odd integer.

Example 2:

Prove by contradiction that if n^2 is even, and then n must be even.

Assumption: There exists a number for n which is odd, but n^2 is even.

So n is odd n = 2k+1

n^2 = (2k+1)^2 = 4k^2 + 4k + 1 (n^2 is odd because any multple of 4 +1 is odd)

So n^2 is odd, this contradicts with the assumption that n is odd and n^2 would be even, this therefore means that the original statement was correct prove my assumption wrong.

Example 3:

Prove by contradiction that √2 is an irrational number.

Assumption: √2 is a rational number

Then √2 = a/b a and b being integers, a and b being in their simplest form

So we can say 2 = a^2/b^2 therefore a^2 = 2b^2 (This means that a^2 is even so a must also be even)

If a is even then we can express it as a = 2n with n being an integer, meaning a^2 = 4n^2

So 4n^2 = 2b^2 (4 and 2 are both divisible by 2 so they both have a common factor of 2)

If a and b are both even then they will have a common factor of 2, this would contradict statement that a and b have no common factor. There fore √2 is irrational

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Proof by exhaustion

For proof by exhaustion you would simply just prove it by doing what is stated in the question, for example if it says show the difference of two consecutive square numbers between 25 and 100 are odd you would simply just do what it asks. Once you have done what is required to prove it YOU MUST ALWAYS ADD A CONCLUDING STATEMENT STATING WHAT YOU HAVE FOUND

Example 1:

Show that the difference of two consectutive square numbers between 25 and 100 are odd.

6^2 = 36 7-6 = 13

7^2 = 49 8-7 = 15

8^2 = 64 9-8 = 17

9^2 = 81 So we can conclude that the difference of two consecutive square numbers is odd

Example 2:

Show that the cube numbers of 3 to 7 are multiples of 9 or 1 more or 1 less than a multiple of 9.

3^3 =27 (9 x 3) So the cube numbers from 3 to 7 are a multiple of 9 or 1 more or 1 less

4^3 = 64 (9 x 7) + 1 than 9 a multiple of 9.

5^3 = 125 (9 x 14) - 1

6^3 = 216 (9 x 24)

7^3 = 343 (9 x 38) + 1

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Proof by deduction

We would have to do proof by deduction algebraically because to do proof by exhaustion can take too long when you are not given a range. WHEN PROVING ALWAYS WRITE A CONCLUSION

Example 1:

Show that the difference of two consecutive square numbers are always odd.

Let the two consecutive numbers be n and n+1 stating n is a positive integer.

So we can now say that (n^2) - (n+1)^2 = n^2 + 2n + 1 - n^2 = 2n + 1

So since 2n is even by adding 1 we have made an odd number. Therefore the difference of two consecutive square numbers is always odd.

Example 2:

Show that all cube numbers squared are multiples of 9 or 1 more or 1 less than a multiple of 9.

Let n be a positive integer,

(3n) = 3, 6, 9 (3n)^2 = 27n^3 - 54n^2 + 36n - 8 = 9 (3n^2- 6n^2 +

(3n-1) = 2, 5, 8 (3n-1)^2 =

(3n-2) = 1, 4, 7 (3n-2)^2 =

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Page 4

Proof by counter example

For proof by counter example we only need one example to disprove a statement.

Example 1

online text book page 152

Example 2

https://www.examsolutions.net/tutorials/proof/using-a-counter-example/?level=A-Level&board=Edexcel&module=Pure-Maths-A-Level&topic=22430

Example 3

https://www.examsolutions.net/tutorials/proof/using-a-counter-example/?level=A-Level&board=Edexcel&module=Pure-Maths-A-Level&topic=22430