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A Levels / Maths

Proof

Master A-Level mathematical proofs and algebraic proof techniques. Learn step-by-step methods for proof by deduction, proof by exhaustion, disproof by counter-example, and classic proof by contradiction including the irrationality of root 2 and the infinity of primes.

A mathematical proof is a formal logical argument that starts with established axioms and proceeds step-by-step to demonstrate a definitive conclusion.
Proof by deduction uses known algebraic rules, identities, and logical deductions to prove that a mathematical statement is always true.
Even numbers are represented algebraically as 2n2n (where nn is an integer), while odd numbers are written in algebraic proofs as 2n+12n + 1 or 2n−12n - 1.
Consecutive integers are represented algebraically as nn, n+1n + 1, and n+2n + 2, allowing properties of sequential numbers and multiples to be proved.
The square of an even number is always even because (2n)2=4n2=2(2n2)(2n)^2 = 4n^2 = 2(2n^2), which is a clear multiple of 22.
The square of an odd number is always odd because (2n+1)2=4n2+4n+1=2(2n2+2n)+1(2n + 1)^2 = 4n^2 + 4n + 1 = 2(2n^2 + 2n) + 1, which has a remainder of 11 when divided by 22.
Proof by exhaustion requires splitting a theorem into all possible individual cases or subsets and verifying that the statement holds true for every single case.
Proof by exhaustion is strictly valid when the number of cases is finite and small enough to test exhaustively, or when exhaustive modular cases cover all integers.
Disproof by counter-example proves a universal mathematical statement false by providing just one specific valid case where the statement does not hold.
Showing examples that work does not constitute a complete mathematical proof, but a single counter-example is sufficient to disprove any universal claim.
Proof by contradiction begins by assuming the exact opposite (negation) of the statement you are trying to prove is true.
In a proof with contradiction, you follow valid logical algebra until you arrive at an impossible result that contradicts established mathematical facts.
Reaching an impossible contradiction proves that your initial negative assumption is false, which confirms that the original statement must be true.
Proving that 2\sqrt{2} is irrational by contradiction starts by assuming 2=ab\sqrt{2} = \frac{a}{b} in simplest coprime form, which inevitably leads to the contradiction that both aa and bb are even.
Proving the infinity of prime numbers by contradiction assumes a finite set of primes exists, then constructs N=(p1×p2×...×pn)+1N = (p_1 \times p_2 \times ... \times p_n) + 1, which leaves a remainder of 11 when divided by any prime on the list.
A-Level Maths Proof Revision: Methods & Contradiction | TSL