Welcome to the World of Proof!

In Mathematics, we don't just "guess" that something is true because it works a few times. We need to be 100% certain. This chapter is all about becoming a mathematical detective. You will learn how to build an unbreakable argument that shows why a statement must be true for every single number, or how to "break" a false statement with just one example. Proof is the foundation of all mathematics—it's how we know the rules we use every day actually work!

Section 1: Speaking the Language of Proof

Before we start building proofs, we need to make sure we are using the right "building blocks." You'll see these terms and symbols throughout your AS Level course.

Important Number Types

  • Integers: These are whole numbers (positive, negative, or zero). Examples: -3, 0, 7, 102.
  • Real Numbers: Pretty much any number you can find on a continuous number line. Examples: 0.5, \(\pi\), \(\sqrt{7}\), -10.
  • Rational Numbers: Numbers that can be written as a fraction \( \frac{a}{b} \) where \(a\) and \(b\) are integers. Examples: \(\frac{1}{2}\), 0.75 (which is \(\frac{3}{4}\)), 5 (which is \(\frac{5}{1}\)).
  • Irrational Numbers: Numbers that cannot be written as a simple fraction. Their decimals go on forever without repeating. Examples: \(\sqrt{2}\), \(\pi\).

The Logic Symbols (Connectives)

Think of these as shorthand for mathematical sentences:

  • \(\equiv\) (Identity/Congruence): This means "is always equal to." For example, \(2(x + 3) \equiv 2x + 6\). No matter what number \(x\) is, both sides are identical.
  • \(\Rightarrow\) (Implication): This means "if... then..." or "implies." Example: You live in London \(\Rightarrow\) You live in England. (The reverse isn't necessarily true!)
  • \(\Leftrightarrow\) (Equivalence): This means "if and only if" (sometimes written as iff). It’s a two-way street. Example: A polygon has three sides \(\Leftrightarrow\) It is a triangle.

Quick Review: Remember that an even number can always be written as \(2n\) and an odd number as \(2n + 1\), where \(n\) is an integer. This is the "secret weapon" for many proofs!

Key Takeaway: Precise language is vital. If a question asks about integers, don't use decimals in your proof!


Section 2: Proof by Deduction

Proof by deduction is the most common method. You start from known facts (assumptions) and use logical steps to reach a conclusion. It's like following a recipe to get to the finished meal.

Step-by-Step: Proving the Sum of Two Odd Numbers is Even

Don't worry if this seems tricky at first! Just follow these logical steps:

1. State your starting point: Let our two odd numbers be \(2m + 1\) and \(2n + 1\) (where \(m\) and \(n\) are integers).
2. Perform the operation: Add them together.
\( (2m + 1) + (2n + 1) = 2m + 2n + 2 \)
3. Show the structure: Factor out a 2 to show it fits the "even" pattern.
\( 2(m + n + 1) \)
4. Conclude: Since \(m, n,\) and \(1\) are integers, their sum is an integer. Any number in the form \(2 \times (\text{integer})\) is even. Proof complete!

Did you know? The letters "Q.E.D." (Quod Erat Demonstrandum) were traditionally written at the end of a proof. It’s Latin for "which was to be demonstrated"—basically a fancy way of saying "I proved it!"

Common Mistake: Don't just use examples (e.g., \(3 + 5 = 8\)). An example shows it works for those numbers, but a deduction proves it works for all numbers.

Key Takeaway: Start with general algebraic expressions (\(n, 2n, 2n+1\)) rather than specific numbers.


Section 3: Proof by Exhaustion

Sometimes, it's impossible to write a general algebraic proof. Instead, you can break the problem down into a few specific cases and test every single one. This is proof by exhaustion—because you are exhausting all possibilities!

Example: Prove that \(n^2 + n\) is even for all integers \(1 \leq n \leq 4\)

Since there are only four numbers to check, we can just test them all:

  • Case \(n = 1\): \(1^2 + 1 = 2\) (Even)
  • Case \(n = 2\): \(2^2 + 2 = 6\) (Even)
  • Case \(n = 3\): \(3^2 + 3 = 12\) (Even)
  • Case \(n = 4\): \(4^2 + 4 = 20\) (Even)

We have checked every possible case in the given range, so the statement is proven.

Using "Even" and "Odd" as Cases

You can also use exhaustion for all integers by splitting them into two cases: Case 1: \(n\) is even and Case 2: \(n\) is odd. If you prove it works for both, you've covered every number in existence!

Analogy: Imagine you want to prove that every door in a corridor is locked. Proof by exhaustion is like walking down the hall and physically turning every single handle.

Key Takeaway: Use exhaustion when there are only a small number of cases to check, or when you can easily split the problem into "even" and "odd" scenarios.


Section 4: Disproof by Counter-Example

Proving something is true for every number is hard. But proving something is false is much easier! To disprove a statement, you only need to find one single case where it doesn't work. This is called a counter-example.

How it works

If someone says: "If \(n\) is a prime number, then \(n\) is always odd," you just need to find one prime number that is even.
Counter-example: \(n = 2\).
2 is a prime number, but it is not odd. Therefore, the statement is disproved.

The "One Strike and You're Out" Rule

In math, a rule must work 100% of the time. If it fails once, the whole statement is false. You don't need to find ten examples of it failing; one is enough.

Quick Review Box:
1. Look at the claim.
2. Try small numbers (\(0, 1, 2\)) or negative numbers to see if you can "break" the rule.
3. State your value of \(x\) and show clearly why the statement fails for that value.

Key Takeaway: A counter-example is the quickest way to destroy a false mathematical claim. Just one "black swan" disproves the claim that "all swans are white."


Final Checklist for Success

  • Do I know the difference between rational and irrational?
  • Can I represent even numbers as \(2n\) and odd numbers as \(2n+1\)?
  • Am I using deduction (algebra) when the statement is true for all numbers?
  • Am I using exhaustion when I can split the problem into a few cases?
  • Am I looking for one single case when asked to disprove?

Don't forget: Always write a small concluding sentence at the end of your proof to explain what you have shown. It makes your argument much stronger!