Mathematical Proof
What Is a Mathematical Proof?
A mathematical proof is a logical argument demonstrating that a statement is always true. It uses previously established results, definitions, and axioms to build a chain of reasoning.
Types of Mathematical Proof
Proof by Deduction
This is the most common form of proof. It involves starting from known facts, definitions, or axioms and using logical steps to arrive at the conclusion.
Example:
Prove that the sum of two even numbers is even.
Let two even numbers be 2a and 2b.
Then their sum is 2a + 2b = 2(a + b), which is divisible by 2, hence even.
Proof by Exhaustion
This method involves checking every possible case individually. It’s only suitable when the number of cases is finite and manageable.
Example:
Check whether for all integers n such that 1 ≤ n ≤ 4, n² ≤ 2n + 4. Show your working.
Check each value of n:
n = 1: 1² = 1 ≤ 6
n = 2: 4 ≤ 8
n = 3: 9 ≤ 10
n = 4: 16 ≤ 12 – is incorrect.
So the statement is not true for all values.
Disproof by Counterexample
To disprove a statement, it’s enough to find one example where it fails.
Example:
Disprove: “All prime numbers are odd.”
Counterexample: 2 is a prime number and it is even.
Proof by Contradiction (A-Level only)
This involves assuming the opposite of what you want to prove and showing that this leads to a contradiction.
Example:
Prove that √2 is irrational.
Assume √2 = a/b where a, b are integers with no common factors.
Then 2 = a²/b² ⇒ a² = 2b².
So a² is even ⇒ a is even ⇒ a = 2k.
Substitute: (2k)² = 2b² ⇒ 4k² = 2b² ⇒ b² = 2k² ⇒ b is even.
So both a and b are even ⇒ they have a common factor of 2 ⇒ contradiction.