Three Different Types Of Proofs

A proof is a structured argument that follows a set up of logical steps. It sets out to evidence if a mathematical statement or conjecture is truthful using mathematical facts or theorems. In one case 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. Substantially, these are things that nosotros assume to be truthful and that we practice not demand to bear witness. Some examples of axioms are:

All multiples of two are even.
Addition is commutative:
Multiplication is commutative:

What must you do in a proof?

The central elements to writing a thorough proof are:

Land any information that yous are using.
Make sure every step logically follows on from the step before.
Make sure all possible cases are covered, eg if y'all are asked to prove for all numbers and you have only proven for odd numbers, and so you have to show for fifty-fifty numbers too.
Stop the proof with a statement.

What are the unlike types of proof?

The different types of proof are defined according to the method being used to do the proof. The principal methods that you lot can find are:

Proof by deduction
Proof past counterexample
Proof by burnout
Proof past contradiction

Proof past deduction

Proof by deduction is the nearly commonly used method of proof, and it involves starting from known facts or theorems, and then going through a logical sequence of steps that show the reasoning that leads you to reach a conclusion that proves the original theorize.

The equation has no real roots. Testify that satisfies the inequality

This is going to involve using the discriminant.

When something has no existent roots, the value of

And so permit'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 become:

Proof Proof by deduction example StudySmarter 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However, when the discriminant formula is no longer valid.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 nosotros substitute in the original equation

This is not possible, so there are no real roots

Therefore as required.

Check out the Proof by Deduction commodity for more than examples.

What about identities?

An identity is a mathematical expression that is always truthful. Information technology is a argument showing that the two sides of the expression are identical. To show an identity , merely manipulate i side of the expression algebraically until information technology matches the other side. A symbol you volition detect 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 tin say that

two. Yous can also exist asked to show trigonometric identities:

Prove that

Consider the diagram beneath:

Proof Pythagorean trigonometric identity StudySmarter Proof of a trigonometric identity, Study Smarter Originals

If nosotros write out trigonometric expressions for and :

Past Pythagoras

And so substituting expressions in for and :

Factoring out :

Divide both sides by (We can practise this considering )

Therefore

Please refer to the Proving an Identity article to expand your knowledge on this topic.

Proof by counterexample

A mathematical statement tin be disproved past finding one counterexample. A counterexample is an example for which a statement is non true. Allow's wait at an case beneath:

Prove that the statement below is not truthful.

The sum of two square numbers is ever a square number.

We tin prove this by counterexample, by finding a single example that proves that the statement is false. So, we demand to discover two square numbers that when added the upshot is non a square number. Let's try 4 and nine.

iv is a square number ( )

ix is a square number ( )

9 + 4 = 13

13 is not a square number.

So the statement is non true.

For more details and examples virtually this type of proof, cheque out the Disproof by Counterexample commodity.

Proof by exhaustion

Proving by exhaustion is done past because every instance possible and checking each example separately.

Prove that the sum of ii sequent square numbers between 1 and 81 is an odd number.

The square numbers between 1 and 81 are:

4, ix, 16, 25, 36, 49, and 64.

Now allow'south use proof by exhaustion, and find these sums.

iv + 9 = thirteen (odd)

ix + 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, take a look at the Proof past Exhaustion article.

Proof by contradiction

Proof by contradiction works slightly unlike. In this instance, in order to show a mathematical statement to be true, you will assume that the opposite of the argument must be faux, and prove that information technology is actually fake.

Testify that there are no integers a and b for which

Presume the opposite: Presume that we tin can find two integers a and b that make the equation true.
If that is the example, and then we tin can divide both sides of the equation by 5:

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 near commonly used method of proof, and it involves starting from known facts or theorems, and so going through a logical sequence of steps to reach a conclusion that proves the original theorize.
Proving identities is done by manipulating one side of the expression algebraically until information technology 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, past assuming that the contrary of the argument must be false, and proving that it is actually imitation.