Proof

Introduction to Proof

• Understand the purpose of mathematical proof, which serves to establish the validity of mathematical statements beyond doubt.
• Familiarise with the language and notation used in writing proofs, such as ‘iff’ (if and only if), ‘∀’ (for all) and ‘∃’ (there exists).
• Appreciate that there are different types of proof, including direct proof, proof by contradiction, and proof by induction.

Direct Proof and Proof by Contradiction

• Grasp the structure of direct proof, which involves assuming the truth of a premise and then deriving a conclusion logically from this premise.
• Understand the principle of proof by contradiction (also known as reductio ad absurdum). This method involves assuming the opposite of what you want to prove, then showing that this assumption leads to an absurdity or contradiction.
• Practice writing your own direct proofs and proofs by contradiction for different mathematical theorems and statements.

Proof by Induction

• Understand the concept of mathematical induction, a method of proof used often when proving statements about natural numbers.
• Learn the two steps of an induction proof: base case (proving the statement is true for an initial value) and the inductive step (showing that if the statement holds for any one natural number, it must also hold for the next).
• Practice applying proof by induction in a variety of contexts, especially sequences and series, inequalities, and divisibility.

Counterexamples and Constructive Proof

• Know that a counterexample can immediately disprove a statement by showing that it does not hold in a specific instance.
• Understand constructive proof, where a mathematical object is shown to exist by actually constructing it.

Remember, this content is meant as a guide to aid your revision journey and should be supplemented with detailed understanding from your learning materials, textbooks, and additional resources. Mastering the art of mathematical proof involves consistent practice.