Proof by exhaustion Revision
A method of proof using algebra to check every possibility, which
leads to a valid proof this is called proof by exhaustion.
Like proof by deduction it is necessary to know how to represent
numbers using notation and algebra.
A refresh on mathematical notation
N => Natural Numbers Positive Integers 1,2,3
Z => Integers Whole Numbers -3,-2,-1,1,2,3
Q => Rational Numbers Numbers that can be without surds
x=p/q when p,q are an element of Z, Q is not equal to 0
R => Real Numbers Numbers that don't include i 3,2/3.-9
C => Complex Numbers All numbers that include i 3-2i
A refresh on representing numbers using algebra
An odd integer can be expressed as
2m + 1
Should be noted that any odd number can be represented as 2n ±
(any odd number)
This can be used for consecutive odd numbers
An even integer can be expressed as
2n
Like for odd numbers you can represent even numbers as 2n ±
(any odd number)
This can be used for consecutive odd numbers
In proof by exhaustion it is not always necessary to represent
numbers like this
And sometimes it also requires you to represent numbers in
different ways based on the question but these representations
are the basics
Worked Example
Prove that 83 is a prime number
A method of proof using algebra to check every possibility, which
leads to a valid proof this is called proof by exhaustion.
Like proof by deduction it is necessary to know how to represent
numbers using notation and algebra.
A refresh on mathematical notation
N => Natural Numbers Positive Integers 1,2,3
Z => Integers Whole Numbers -3,-2,-1,1,2,3
Q => Rational Numbers Numbers that can be without surds
x=p/q when p,q are an element of Z, Q is not equal to 0
R => Real Numbers Numbers that don't include i 3,2/3.-9
C => Complex Numbers All numbers that include i 3-2i
A refresh on representing numbers using algebra
An odd integer can be expressed as
2m + 1
Should be noted that any odd number can be represented as 2n ±
(any odd number)
This can be used for consecutive odd numbers
An even integer can be expressed as
2n
Like for odd numbers you can represent even numbers as 2n ±
(any odd number)
This can be used for consecutive odd numbers
In proof by exhaustion it is not always necessary to represent
numbers like this
And sometimes it also requires you to represent numbers in
different ways based on the question but these representations
are the basics
Worked Example
Prove that 83 is a prime number
