Proofs
Information
Proofs are used in mathematics in order to shorten the time it takes to complete a future operation as by proving a method works to other mathematicians allows it to be widely used and can save everyone alot of time if it is a shorter method
Below is a simple prrof for finding if everything multiplied by 2 is an even number
First we can define a natuaral numbers as n
Therefore doubling n gives us 2n which is divisible by 2 therefore for n this is correct
Next we do the same operation for n+1 to see if the next number after n is also divisible by 2
2(n+1) = 2n + 2
This is also divisible by 2 and therefore the statement is also true for n+1
Finnaly we do this operation for 1
2×1 = 2
This is divisible by 2 as well and therefore we can state:
For all Natural Numbers multiplying by 2 gives an even number
Steps for a proof
First we see if true for a value of 1
Then we see if true for a value of a constant (e.g.n or k)
Next we check for the constant plus 1 (e.g.[n+1] or [k+1])
Finally we write a paragraph about what we have proved and how we have proved it
Example 1
Prove:
Assume n=1:
Therefore this is true for n = 1 as both equations have the same answer
Now assume true for n = k
Now consider for k+1:
As we assumed that for n=k:
We can rewrite the equation as:
Now for the other equation:
We once again have the same answer from both equations for k+1 and it is therefore true for k+1 as well
(Must Right This)
But this is given the result with k+1 replacing k
Therefore, if the result is true for n=k it is also true for n = k+1
Since it is true for n=1
By induction it is therefore true for all positive integers n
Example 2
Prove that for all positive integers n,
First we test for n=1
This is true and therefore the statement is true for n=1
Now we assume this is true for n=k
Letting n=k+1
Replacing k with k+1 in:
This is the same value we got in the other equation therefore the statement is true for k+1
But this is given the result with k+1 replacing k
Therefore, if the result is true for n=k it is also true for n = k+1
Since it is true for n=1
By induction it is therefore true for all positive integers n
Example 3
Prove that for all positive integers n:
Letting n=1
This is true and therefore the statement is true for n=1
Now we assume this is true for n=k
Letting n=k+1
Taking common parts outside the brackets
Replacing k with k+1 in:
This is the same value we got in the other equation therefore the statement is true for k+1
But this is given the result with k+1 replacing k
Therefore, if the result is true for n=k it is also true for n = k+1
Since it is true for n=1
By induction it is therefore true for all positive integers n
Example 4
Now we will do a different kind of proof where we are proving we can find a number in a sequence using an equation:
Given:
Prove:
First we test for n=1
First we test for n=1
Now testing:
This is true and therefore the statement is true for n=1
Now we assume this is true for n=k
and
Letting n=k+1
Replacing k with k+1 in:
Using the value of
we have already found:
Now we need to remember that
and therefore
Employing this knowledge to set the power to k in our equation from k-1 gives:
Now we have the same value that we got previously and therefore the statement is true for k+1
But this is given the result with k+1 replacing k
Therefore, if the result is true for n=k it is also true for n = k+1
Since it is true for n=1
By induction it is therefore true for all positive integers n
Example 5
Prove
is divisible by 7 for all values n ≥ 1
First we let n=1
Now we need to check that 35 is divisible by 7
This is divisible by 7 therefore this statement is true for n=1
Now we assume this is true for n=k
Where m is any integer value (Therefore 7m is always divisible by 7)
We can rearrange this to:
Letting n=k+1
Replacing
with
from the equation derived above gives us:
(This can be done with the
as well if you rearrange the equation differently)
This is a multiple of 7 and is therefore divisible by 7 meaning that the expression is true for n=k+1
But this is given the result with k+1 replacing k
Therefore, if the result is true for n=k it is also true for n = k+1
Since it is true for n=1
By induction it is therefore true for all positive integers n
Example 6
This final example shows a proof for a matrix
For
,Prove that
Letting n=1
This is the same value we have for A and therefore it is true for n=1
Now we assume this is true for n=k
Letting n=k+1
We know that
,Therefore using this multiplication we can see if we get the same equation for
This is the same expression we found for n=k+1 meaning that the expression is true for n=k+1
But this is given the result with k+1 replacing k
Therefore, if the result is true for n=k it is also true for n = k+1
Since it is true for n=1
By induction it is therefore true for all positive integers n