A proof is a structured argument that follows a set of logical steps. It sets out to prove if a mathematical statement or conjecture is true using mathematical facts or theorems. Once a conjecture has been proved, it becomes a theorem . An example of a theorem is the fact that an even Number squared is even.
Theorems are based on axioms. Axioms are defined as a statement or proposition on which a structure is based. Essentially, these are things that we assume to be true and that we do not need to prove. Some examples of axioms are:
All multiples of 2 are even.
Addition is commutative: \(a + b = b + a\)
Multiplication is commutative: \(a \cdot b = b \cdot a\)
What must you do in a proof?
The key elements to writing a thorough proof are:
State any information that you are using.
Make sure every step logically follows on from the step before.
Make sure all possible cases are covered, eg if you are asked to prove for all numbers and you have only proven for odd numbers, then you have to prove for even numbers too.
Finish the proof with a statement.
What are the different types of proof?
The different types of proof are defined according to the method being used to do the proof. The main methods that you can find are:
Proof by deduction
Proof by Deduction is the most commonly used method of proof, and it involves starting from known facts or theorems, then going through a logical sequence of steps that show the reasoning that leads you to reach a conclusion that proves the original conjecture.
The equation id="2904200" role="math" \( kx^2 - 2kx + 4 = 0 \) has no real roots. Prove that \(k\) satisfies the inequalityid="2904204" role="math" \(0 \leq k < 4\)
This is going to involve using the discriminant.
When something has no real roots, the value of \(b^{2} - 4 \cdot a \cdot c < 0\)So let's just substitute values of \(a\), \(b\) and \( c \).
\(a = k, \; b = -2k\) other \(c = 4\)\((-2k)^2 - 4(k)(4) = 4k^2 - 16k\)So \(4k^2 - 16k < 0\), as this has no real roots, the value of the discriminant has got to be less than 0.
\(k(4k-16)<0\)
So if we sketch this out, we get:
Proof by deduction example, Marilu García De Taylor - StudySmarter Originals
You can see in the graph that \(k(4k-16)<0\)when the curve is below the x-axis . This happens when\(0 < k < 4\)
However, when \(k = 0\) the discriminant formula is no longer valid.
If we substitute \(k = 0\) in the original equation\(kx^2-2kx+4=0\)\((0)x^2 - 2(0)x + 4 = 0\) \(4 = 0\)This is not possible, so there are no real roots
Therefore \(0 \leq k < 4\) as required.
Check out the Proof by Deduction article for more examples.
What about identities?
An identity is a mathematical expression that is always true. It is a statement showing that the two sides of the expression are identical. To prove an identity , simply manipulate one side of the expression algebraically until it matches the other side. A symbol you will find in identities is ≡, which means 'is always equal to'. Here are a couple of examples:
1. Prove that \((2x + 3)(x + 4)(x - 1) = 2x^3 + 9x^2 + x - 12\)
Expand the brackets on the left-hand side of the identity and combine like terms
\((2x + 3)(x + 4)(x - 1) = (2x + 3)(x^{2} - x + 4x - 4)\) \(= (2x + 3)(x^2 + 3x - 4)\) \(= 2x^3 + 6x^2 - 8x + 3x^2 + 9x - 12\) \(= 2x^3 + 9x^2 + x - 12\)
Therefore, we can say that \((2x + 3)(x + 4)(x - 1) \equiv 2x^3 + 9x^2 + x - 12\)
Proof of a trigonometric identity, Study Smarter Originals
![]()
If we write out trigonometric expressions for \( a \) and \( b \):
\(a = c \cdot \sin \theta\) \(b = c \cdot \cos \theta\)
By Pythagoras \(a^2 + b^2 = c^2\)
So substituting expressions in for \(a\) and \(b\):\((c \cdot \sin{\theta})^2 + (c \cdot \cos{\theta})^2 = c^2 \cdot \sin^2{\theta} + c^2 \cdot cos^2{\theta}\) \(c^2 \sin^2{\theta} + c^2 \cdot \cos^2{\theta} = c^2\)
Factoring out \(c^2\):
\(c^2(\sin^2 \theta + \cos^2 \theta) = c^2\)Divide both sides by \(c^2\) (We can do this because \(c \neq 0\))
Therefore \(\sin^{2}\theta + \cos^{2}\theta = 1\)
Proof by counterexample
A mathematical statement can be disproved by finding one counterexample. A counterexample is an example for which a statement is not true. Let's look at an example below:
Prove that the statement below is not true.
The sum of two square numbers is always a square Number.
We can prove this by counterexample, by finding a single example that proves that the statement is false. So, we need to find two square numbers that when added the result is not a square number. Let's try 4 and 9.
4 is a square number ( \(2^{2}\))
9 is a square number ( \(3^{2}\))
9 + 4 = 13
13 is not a square number.
So the statement is not true.
For more details and examples about this type of proof, check out the Disproof by Counterexample article.
Proof by exhaustion
Proving by exhaustion is done by considering every example possible and checking each case separately.
Prove that the sum of two consecutive square numbers between 1 and 81 is an odd number.
- The square numbers between 1 and 81 are:
4, 9, 16, 25, 36, 49, and 64.
4 + 9 = 13 (odd)
9 + 16 = 25 (odd)
16 + 25 = 41 (odd)
25 + 36 = 61 (odd)
36 + 49 = 85 (odd)
49 + 64 = 113 (odd)
All these numbers are odd, so the statement has been proved.
For more examples, have a look at the Proof by Exhaustion article.
Proof by contradiction
Proof by Contradiction works slightly different. In this case, in order to prove a mathematical statement to be true, you will assume that the opposite of the statement must be false, and prove that it is actually false.
Prove that there are no Integers a and b for which \(5a + 10b = 1\)
- Assume the opposite: Assume that we can find two Integers a and b that make the equation \(5a + 10b = 1\)true.
- If that is the case, then we can divide both sides of the equation by 5:
\(\frac{5}{5} \cdot a + \frac{10}{5} \cdot b = \frac{1}{5}\) \( a + 2 \cdot b = \frac{1}{5} \)If a and b are Integers, then the result of \(a + 2b\) must be an integer too, therefore \(a + 2b\)cannot result in the fraction \(\frac{1}{5}\), which is what the equation states. Here we have a contradiction , which makes our assumption false.
- As we have proved the opposite statement to be false, the original statement is proved to be true. Therefore, we can say that the statement "There are no integers a and b for which \(5a + 10b = 1\)" is true.
To find out more about this type of proof, follow the link to the Proof by Contradiction article.
Proof-Key takeaways
A proof is a sequence of logical steps used to prove a mathematical statement or conjecture.
Proof by deduction is the most commonly used method of proof, and it involves starting from known facts or theorems, then going through a logical sequence of steps to reach a conclusion that proves the original conjecture.
Proving identities is done by manipulating one side of the expression algebraically until it matches the other side.
Proof by counterexample is done by using a counterexample to prove that a statement is not true.
Proof by exhaustion is done by considering all possible cases and proving each case separately.
Proof by contradiction proves a mathematical statement to be true, by assuming that the opposite of the statement must be false, and proving that it is actually false.
Explanations 
Exams 
Magazine 
