Pre-U Further Mathematics CAIE

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.