Summary Proof and Disproof

Proof and Disproof
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Disproof by “counter example”
Possible to use an example to prove that something is not always true… called a counter
example

Example:
Disprove by counter example that ( x +1)2 ≡ x 2 + 1 for all x
Try x = 0 1=1
x=1 4≠4
x=2 Therefore x = 1 is a counter example to ( x +1)2 ≠ x 2 + 1 for all x

Trial and Error
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Proof by deduction
A series of logical steps to reach the required solution
Use algebra
Useful in the algebra to denote even numbers as 2n consecutive odd number as 2n+1, 2n-1
or 2m+1, 2m-1

Example:
Prove the product of an even number add an odd number is always even
Let even = 2n
Let odd = 2m+1
Product = 2n(2m+1)
= 4nm+2n
= 2(nm+n) ¿>¿ Product is an even number
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Proof by exhaustion
Considering some examples do not substitute a mathematical proof… however… where it is
possible to ‘check’ all possibilities this can lead to a valid proof

Example:
Prove that 13 is a prime number
13 is not divisible by 2, 3, 5, 7, 11 so 13 must be a prime number
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