A Levels
Maths topic guides
Explore UK Maths study resources with topic guides, key knowledge, exam-style questions, and revision support for GCSE and A-level students.
Introduction to Proof by Deduction
Proof by deduction is a method of proving a mathematical statement by starting from known facts, definitions, identities, or previously established results and using logical steps to reach a definite conclusion. It is widely used in algebra, number theory and geometry. In A Level Mathematics, students should be able to construct clear deductive arguments, justify every step, and distinguish deduction from other forms of reasoning such as induction or checking individual examples.
Open topic02Mathematical Proof by Contradiction
Mathematical proof by contradiction is a method used to prove that a statement is true by assuming that it is false and then showing that this assumption leads to an impossible or contradictory result. This method is widely used in A Level Maths, particularly in proofs involving irrational numbers, divisibility, prime numbers and number properties. Students should understand how to form the opposite assumption correctly, develop a logical argument, identify the contradiction and clearly state the final conclusion.
Open topic03Proof by Exhaustion and Counter Examples
Proof by exhaustion and counterexamples are important methods used in mathematical reasoning. Proof by exhaustion establishes that a statement is true by checking every possible case within a finite set, while a counterexample disproves a universal statement by finding just one case where it fails. In A Level Maths, students need to understand when proof by exhaustion is valid, how to organise cases systematically, and how counterexamples can be used to show that mathematical claims are false.
Open topic