💡 The Big Idea
A proof is a logical journey from what you know to what you want to show. It's how mathematicians guarantee that something is true for every number in a set, not just a few examples. There are four main ways to do this:
Proof by Deduction
Use algebra and logic to show a statement is true for all numbers in a set.
Proof by Exhaustion
Check every allowed value. Tedious, but sometimes necessary for finite cases.
Disproof by Counter Example
Find just one value that breaks the claim to disprove it completely.
Language & Notation
Integers (ℤ), sets (ℕ, ℚ, ℝ), and conventions like 2n for even numbers.
📖 Language of Proof
What Does It Mean to Prove Something?
In everyday life, "proof" might mean strong evidence. In maths, it's stricter: a proof is a series of logical steps that shows whether a statement is definitely true or false for a clearly defined set of numbers.
If you say "the sum of any three odd numbers is odd," you can't just try three examples and declare victory. You need to prove it works for all odd numbers. That's where these techniques come in.
Key Notation You'll Use
Integers: ℤ
The set of all whole numbers: ..., -2, -1, 0, 1, 2, ... (no fractions, no decimals)
Left-Hand Side (LHS) & Right-Hand Side (RHS)
In an equation like 6n − 3 = 3(2n − 1), the part before = is LHS; the part after is RHS. Proving they're equal is a core tactic.
The Major Sets of Numbers
Proofs often specify which set of numbers they apply to. Know these symbols:
ℕ — Natural Numbers
1, 2, 3, ... (positive whole numbers, sometimes includes 0)
ℤ — Integers
..., -2, -1, 0, 1, 2, ... (all whole numbers)
ℚ — Rationals
Any number you can write as a fraction: 1/2, -3/4, 5, etc.
ℝ — Real Numbers
Everything: integers, fractions, irrationals like π and √2
Conventions for Describing Numbers
When you need to work with even or odd numbers (or multiples in general), use these standard forms so examiners instantly understand what you mean:
Even Numbers
2n where n is any integer. Examples: 2(1) = 2, 2(5) = 10, 2(-3) = -6
Odd Numbers
2n + 1 or 2n − 1 where n is any integer. Examples: 2(2) + 1 = 5, 2(0) + 1 = 1, 2(3) − 1 = 5
Three Consecutive Odd Numbers
2n − 3, 2n − 1, 2n + 1 — note we start with the middle value and work outwards.
Multiples of k
kn where n is any integer. "Multiple of 5" → 5n, "Multiple of 7" → 7n
📝 Proof by Deduction
What Is It?
Proof by deduction is when you use algebraic manipulation and logical reasoning to show a statement is true for all numbers in a set. You set up a general expression, simplify it, and show it has the property you claimed.
Unlike proof by exhaustion (which checks individual cases), deduction works once for the entire set — it's the most powerful technique for infinite sets.
How to Prove by Deduction: The Method
- Define general numbers with "Let" — Start with a statement like "Let n be any integer" or "Let the three consecutive odd numbers be 2n − 1, 2n + 1, 2n + 3"
- Apply the operation — Add them, square them, whatever the problem asks.
- Simplify algebraically — Combine like terms, factor if helpful.
- Recognise the pattern
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