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# Proof|Definition & Meaning

## Definition

The proof of a **mathematical** statement is a series of **logical**, mathematical **arguments** that verify its validity and **truthfulness**. In math, the proof of any statement involves **axioms** and proven **theorems**, often from the particular branch of mathematics where the **statement** appears.

**Figure 1** shows the **proof** for the statement: “If two **angles** are linear pairs, then they are **supplementary** angles.”

## History

The word “**proof**” is derived from the Latin word “**probare**,” meaning “to test.” It also has its origin in the Spanish word “**probar**,” the Italian word “**provare**,” and the German word “probieren,” which all means “to try.”

The idea and **demonstration** of mathematical **proof** were first presented in ancient **Greek** mathematics. Thales and Hippocrates gave the first proofs of the fundamental theorems in geometry.

The **axiomatic** method given by **Euclid** revolutionized mathematical proof. The procedure starts with **axioms**, undefined terms, and **propositions** self-evidently assumed **true** and proved using **deductive** logic.

## Purpose

The purpose of formulating **proof** is to convince an audience of the **truth** of a statement. Many rigorous **arguments** are made to prove the **authenticity** of the statement.

There are many **ways** to present **proof **through different **premises**. A vague or **incomplete** argument does not gain **acceptance** of the proof.

## Mathematical Logic

The field of **mathematical** logic forms the basis for the **concept** of proof. It studies** formal** mathematical logic with sub-areas such as set theory, **proof** theory, and **model** theory.

### Proof Theory

The proof **theory** involves mathematical **techniques** to analyze proofs represented as mathematical **objects**. Proofs are presented as **data structures** such as boxed lists, lists, and **trees** which are inductively defined and constructed according to **rules** of inference and **axioms**.

The proof theory includes **ordinal** analysis, reverse mathematics, **structural** proof theory, **provability** logic, automated **theorem** proving, proof complexity, and proof **mining**.

## Conjecture

A conjecture is a **tentative** statement or **proposition** made without any proof. It is an idea observed when viewing a similar **pattern** that holds for many different **cases**. It is a conclusion that is not **rigorously** proved.

A conjecture can also be called a “**hypothesis,**” but it is not written in a **testable** or **formal** way. An example of a conjecture is** Riemen’s** Hypothesis which paved the way for many areas of **mathematics** to prove them.

## Axiom

An axiom is a **statement** that is considered to be **true** to establish it as a **starting** point for further **arguments** and reasoning. It is also known as an assumption or **postulate**. It is derived from the Greek word “**axioma**,” which means “something that is considered **worthy** or fit” or “something that proposes itself as **evident.**”

For example, the **symmetric** axiom states that if **x = z,** then **z = x**, also known as the axiom of **equality**.

## Theorem

A theorem is a **statement** that has already been **proved** through deductive **logic** arguments. It is a **conjecture** that has been verified using **axioms** to start logical reasoning and is accepted **worldwide**. An example of a theorem is **Fermat’s Last **Theorem which Andrew Wiles proved in 1995.

## Types of Proof

There are various **methods** to prove a **statement** which are given as follows:

### Direct Proof

In **direct** proof, a statement is proved by **logically** using mathematical definitions, **axioms**, and known theorems. **Figure 2** shows the direct proof that the sum of two **odd** integers is always an **even** integer.

### Proof by Contradiction

In proof by contradiction, a statement **opposite** to the **original** statement is proved. In the proof, a **contradiction **proves the opposing statement to be **false, **thereby proving the original statement **true**.

For example, to **prove** by contradiction that if (m^{3} + 5) is **odd**, then **m** is **even** for all integers, it is assumed that **m** is **odd,** which **negates** the given proposition. **Figure 3** shows its proof by using the **contradiction** method.

The last **equation** in figure 3 is impossible (**contradiction**) as **5/2 **(rational number) is not an **integer,** whereas the right-hand side results in an integer under the **closure** property w.r.t **addition. T**herefore the assumption that** m** is **odd** must be **false** for (m^{3} + 5) to be odd. Hence,** m** must be **even**.

### Proof by Contraposition

Proof by contraposition refers to proving the **contrapositive** statement “if not **y,** then not **x,**” which proves the **original** statement “if **x,** then **y**.”

For example, to **prove** the statement “if a^{2} is **even**, then the integer **a** is even” by contraposition means proving the **equivalent** assumption “if **a** is not **even**, then a^{2} is **not** even.”

Suppose **a** is not even, then it is **odd**. By definition, the **product** of two odd integers is **odd**, so a.a = a^{2} is odd, which means a^{2} is **not even**. Hence, by contraposition, it is **proved** that if a^{2} is** even**, then** a** is even.

### Proof by Induction

Proof by induction requires proving a **base case** by inductive arguments. If the base case **P(n)** and the following case, e.g., **P(n + 1),** is provable, then the other **infinite** cases are also provable. For example, to prove by **induction** that the positive integer (2j – 1) represents an **odd** number, consider **figure 4**.

## Example

Prove that the **sum** of any two **consecutive** numbers is an **odd** number using direct **proof**, proof by contradiction, and contraposition.

### Solution

#### Direct Proof

Assume **g** and **h** are two **consecutive** integers, so by **definition** of consecutive numbers:

h = g + 1

Adding** g** and** h** and placing the value of **h** in the **equation** gives:

g + h = g + (g + 1)

g + h = 2g + 1

As **(2g + 1)** represents an **odd** number, hence the **sum** of two **consecutive** integers is an **odd** number.

#### Proof by Contradiction

Assume **g** and **h** are consecutive integers and the sum of **g** and** h** is not odd. If (g + h) is **not odd**, then there **exists** no integer **l** such that:

g + h = 2l + 1

But as **g** and **h** are **consecutive**, then:

g + h = g + (g + 1) = 2g + 1

Hence, there is a **contradiction** between** (g + h) = 2l + 1** since there exists no integer **l** and **(g + h) = 2g + 1** as the integer **g** exists, so:

g + h ≠ 2l + 1

Hence, the **assumption** that the sum of **g** and **h** is not odd must be **false**, so it is proved by contradiction that the sum of two **consecutive** numbers is **odd**.

#### Proof by Contraposition

The **contrapositive** statement will be if the sum of **g** and **h** is not odd, then **g** and **h** are not consecutive integers. If **(g + h)** is not odd, then there does not exist an integer** w** such that:

g + h = 2w + 1

So, **g + h = a + (a + 1)** does not hold for any integer **a**. But since **(a + 1)** is the successor of **a**, it implies that **g** and **h** cannot be consecutive. Hence, the **contrapositive** statement is **proved,** ultimately proving the **original** statement.

*All the images are created using GeoGebra.*