Further Mathematics

Pure Mathematics

# Proof

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Proof

Proof

Understanding Proof

- The process of
**proof**is a logical methodology applied in mathematics to verify the truth of an existing statement or theory. - A mathematical proof is a
**deductive argument**for a mathematical statement which, theoretically, could be checked by a competent mathematician. - Using logic and assumptions, we can identify and systematically verify every operation.
- The primary goal of proof is to establish
**absolute truth**, independent of individual beliefs.

Direct Proof

- A
**direct proof**is the simplest form of proof, where you start at the premises and use logical operations to reach the conclusion. - Direct proofs most often use the properties of mathematics such as the properties of equality or definitions or theorems.
- Direct proofs make use of
**simple, clear and logical steps**that effectively lead to the proposition in question. - Establishing a series of these interconnected certainties can provide an absolute verification of the statement.

Indirect Proof

- An
**indirect proof**, also known as proof by contradiction, begins with an assumption that is the opposite of the statement to be proven. - If this assumption leads to a contradiction, then the original statement must be correct.
**Contradiction**is a powerful tool in indirect proof. It allows us to assume the reverse of our desired outcome, and then to demonstrate that this assumption inevitably leads to a logical inconsistency.

Existence Proof

- An
**existence proof**demonstrates that at least one instance of a particular statement is true. - This type of proof often uses construction or contradiction to prove that a mathematical entity with a certain property exists.
- Existence proofs are often less specific, as they don't necessarily provide a method for finding the entity, just proving that it
**exists**.

Uniqueness Proof

- A
**uniqueness proof**shows that only one instance of a particular mathematical statement is true. - Often used in geometry or calculus, these proofs usually assume that two objects exist and are the same, leading to a contradiction.
- Ultimately, uniqueness proofs establish that a
**single, unique solution**exists, which solidifies the statement's validity.

Proof by Induction

- Proof by induction is a method of proof in mathematics where a statement is shown to be true for all natural numbers.
- The process involves establishing the
**truth of a statement for an initial value**(commonly 1 or 0) and then proving that if it holds true for a certain case, then it also holds true for the next immediate case. - This method is particularly useful for proving statements about
**sequences**or**series**and often used with inequalities.

Mathematic Notations in Proof

- Use of maths notations in proofs can vastly improve the
**clarity**and**conciseness**. - Familiarising yourself with these notations — such as equalities, inequalities, summation, product notation, etc. — is critical to understanding and constructing mathematical proofs.