Fundamentals of Proof

A proof is a logical argument demonstrating that a certain proposition or statement is true, it involves assumptions, propositions, intermediate conclusions and a final conclusion.
Direct proof is the most common type of proof. Here, you start from a given set of assumptions and use them to show that a certain conclusion is valid.
Proof by contradiction (also known as reductio ad absurdum) involves assuming that the statement you’re trying to prove is not true, and then showing that this leads to a contradiction.
Proof by induction is typically used for proving statements involving positive integers. It is characterised by two steps: the base case and the inductive step.
In a base case, you prove the statement for a specific small number, typically 1.
In the inductive step, you assume the statement holds for an arbitrary positive integer ‘k’. Then you show that, under that assumption, it must also work for ‘k+1’.
Proof by exhaustion involves checking all the possible cases in a finite set. This method is useful when the number of cases is relatively small.

Types of Statements and Definitions

Conditional statements have a hypothesis and conclusion. They follow the structure “if p, then q.”
Biconditional statements are based on “if and only if” and imply that the hypothesis and conclusion depend on each other. They are true in both directions.
Counterexamples are used to disprove conditional and biconditional statements. They are examples that conform to the initial conditions of the statement but don’t produce the correct outcome.
An axiom or postulate is a statement that is accepted without proof. They form the basis of any mathematical theory.
A theorem is a major result that has been proved to be true using axioms or other already established theorems.
A lemma is a “helping theorem,” a proposition that isn’t interesting in its own right, but is used to assist in the proof of a larger theorem.
A corollary is an immediate consequence of a theorem.

Proof Techniques

Modus Ponens is a deductive argument meaning ‘the mode of affirming’. If a statement of “if p then q” is true and p is true, then q must also be true.
Modus Tollens is another deductive argument, meaning ‘the mode of denying’. This says if “if p then q” is true and q is false, then p must be false.
Disjunctive Syllogism also referred to as ‘the either/or scenario’. Given the statement “p or q”, if ‘p’ is false then ‘q’ must be true and vice-versa.
The use of Quantifiers in mathematics, such as “for all” and “there exists”, allows for more generalised and powerful forms of statements.
Familiarity with Logic and Set Notation will facilitate the reading and writing of formal mathematical proofs.

Remember that understanding and practising the construction of mathematical proofs is a critical skill in further mathematics. Being able to follow a series of logical steps to arrive at a conclusion, and then to articulate that process in a clear and concise way, is a significant part of your maths development.