Proof (A2 Unit 3: Pure Mathematics B) — WJEC A-Level Mathematics
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Proof (A2 Unit 3: Pure Mathematics B) explained
Proof by contradiction assumes the logical negation of the proposition to be proved and demonstrates that this hypothesis inescapably yields a mathematical contradiction.
Read the full explanation
To prove √2 is irrational, assume √2 = a/b where a, b ∈ ℤ share no common factors (in simplest form). Squaring gives 2 = a²/b², so a² = 2b², meaning a² is even, which necessitates that a is even (a = 2k). Substituting gives 4k² = 2b², so b² = 2k², implying b is also even. This contradicts the assumption that a and b have no common factors; hence, √2 must be irrational. To prove the infinity of primes, assume a finite set of primes {p₁, p₂, ..., p_n} and construct N = (p₁ p₂ ... p_n) + 1; dividing N by any prime in the list leaves remainder 1, implying either N is prime or has a prime factor not in the list.
Your focus
- Structure and write rigorous proofs by contradiction for unfamiliar mathematical statements.
- Prove the irrationality of √2 using divisibility and parity arguments.
- Prove the infinity of primes using Euclid's construction and proof by contradiction.
Proof (A2 Unit 3: Pure Mathematics B) exam tips
Marking Points
- stating the initial assumption of the negation of the statement to be proved
- rigorous algebraic manipulation displaying an impossible condition (such as common factor of 2)
- explicitly identifying the contradiction with the original assumption
- concluding that the initial proposition must therefore be true
Examiner Tips
- 💡Start clearly with: 'Assume the contrary, that √2 is rational and can be written as a/b in simplest form'.
- 💡Conclude with: 'This contradicts the assumption that a and b have no common factors, so √2 is irrational'.
Common Mistakes
- forgetting to specify that the fraction a/b is in its simplest irreducible form with no common factors
- stopping after showing a is even without continuing to prove that b must also be even