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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: 
Multiplication is commutative: 
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.
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 counterexample
Proof by exhaustion
Proof by contradiction
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 
has no real roots. Prove that 
satisfies the inequality
This is going to involve using the discriminant.
When something has no real roots, the value of 
So let's just substitute values of 
, 
and 
.
other 
So 
, as this has no real roots, the value of the discriminant has got to be less than 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 
when the curve is below the x-axis . This happens when
However, when 
the discriminant formula is no longer valid.
If we substitute 
in the original equation
This is not possible, so there are no real roots
Therefore 
as required.
Check out the Proof by Deduction article for more examples.
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 
Expand the brackets on the left-hand side of the identity and combine like terms
Therefore, we can say that
2. You can also be asked to prove trigonometric identities:
Prove that 
Consider the diagram below:
Proof of a trigonometric identity, Study Smarter Originals
If we write out trigonometric expressions for 
and 
:
By Pythagoras 
So substituting expressions in for 
and 
:
Factoring out 
:
Divide both sides by 
(We can do this because 
)
Therefore 
Please refer to the Proving an Identity article to expand your knowledge on this topic.
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 ( 
)
9 is a square number ( 
)
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.
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.
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 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 
true. 
must be an integer too, therefore 
cannot result in the fraction 
, which is what the equation states. Here we have a contradiction , which makes our assumption false.
" is true. To find out more about this type of proof, follow the link to the Proof by Contradiction article.
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.
To write a proof in Maths, start with theorems and axioms before performing mathematical processes, and finally finish with a statement concluding your proof.
A mathematical proof is a structured argument that follows a sequence of logical steps using facts and theorems to prove if a mathematical statement is true. It shows the reasoning behind every step and culminates with a final statement.
Proof gives us evidence for our statements and the certainty that what we are using is accurate.
The three main types of proof are proof by deduction, by counterexample, and by exhaustion. Another important method of proof studied at A-levels is proof by contradiction.
Question
Prove this statement is not true.
for all real
.
Show answer
Answer
Let x = 3.
So the statement is untrue.
Show question
Question
Prove this statement is untrue.
All odd numbers squared are even.
Show answer
Answer
Choose an odd number such as 5.
. This is not even, so the statement is untrue.
Show question
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