# Proof

# 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.