Introduction to Proof and Mathematical Argument

Welcome to one of the most important chapters in your A2 Pure Maths journey! While you have been "doing" maths for years, proof is where you learn how to "justify" it. In Unit P2, we move beyond just finding the answer; we focus on constructing rigorous, bulletproof arguments that leave no room for doubt.

Think of a mathematical proof like a trial in a courtroom. You cannot just say something is true because it looks right; you must provide evidence and logical steps that lead inevitably to your conclusion. Whether you are aiming for an A* or just trying to pass, mastering the language of logic will help you pick up marks for "Quality of Written Communication" and AO2 (Mathematical Argument) objectives.

1. The Language of Logic

To write a good proof, you need to use the right "connecting" words. Using these correctly shows the examiner that you understand the flow of logic.

Key Terms and Symbols

  • Therefore (\(\therefore\)): Used when one statement follows logically from the one before it.
  • Identically Equals (\(\equiv\)): Used for identities. An identity is true for all values of the variable. For example, \((x + 1)^2 \equiv x^2 + 2x + 1\). This is different from a normal equals sign (\(=\)), which is usually only true for specific values (like \(x = 3\)).
  • Implies (\(\implies\)): This symbol means "if the first part is true, then the second part must also be true." For example, \(x = 3 \implies x^2 = 9\).
  • Is Implied By (\(\impliedby\)): This means the logic works backwards. For example, \(x^2 = 9 \impliedby x = 3\). (Note: It doesn't work the other way because \(x^2 = 9\) could also mean \(x = -3\)).
  • If and Only If / Equivalence (\(\iff\)): This is used when the logic works in both directions. For example, \(x + 1 = 4 \iff x = 3\).

Necessary vs. Sufficient Conditions

These two terms are often confused, but they have very specific meanings:

1. A Necessary condition is something that must be true for a result to happen. Without it, the result is impossible.
Example: Being 18 is necessary to vote, but it isn't enough on its own (you also need to register!).

2. A Sufficient condition is something that is enough to guarantee a result.
Example: Being a dog is sufficient to say you are an animal. It’s enough information to reach the conclusion.

2. Proof by Logical Deduction

This is the most common form of proof. You start from known facts or definitions and use a chain of logical steps to reach a conclusion. For Unit P2, you are expected to handle these even in unstructured forms (where the question doesn't tell you exactly what steps to take).

Step-by-step approach:

  1. State your starting point clearly (e.g., "Let \(n\) be an even integer, so \(n = 2k\)").
  2. Perform algebraic manipulations.
  3. Show every step of your working.
  4. Finish with a concluding statement that relates back to the original question.

Quick Tip: When proving things about odd or even numbers, always use:
- Even numbers: \(2n\)
- Odd numbers: \(2n + 1\) (or \(2n - 1\))

3. Proof by Contradiction

This is a powerful "indirect" method of proof added in the A2 syllabus. Instead of proving something is true, you assume it is false and show that this leads to a total "mathematical disaster" (a contradiction).

The Process:

1. Assume the opposite: If you want to prove statement \(P\) is true, start by assuming \(P\) is false.

2. Logical steps: Use algebra or logic based on that assumption.

3. The Crash: You will eventually reach a statement that is clearly impossible (e.g., \(1 = 0\) or saying a number is both even and odd).

4. Conclusion: Since your assumption led to nonsense, the original statement must be true.

Common Example: Proving \(\sqrt{2}\) is irrational.
You would start by assuming \(\sqrt{2}\) is rational (meaning it can be written as a fraction \(\frac{a}{b}\)). After some algebra, you would show that \(a\) and \(b\) must both be even, meaning the fraction was never in its simplest form. This contradicts the definition of a rational fraction, so \(\sqrt{2}\) must be irrational.

Key Takeaway: If assuming the "not" version of a statement leads to a contradiction, the original statement is proven!

4. Disproof by Counter-Example

In mathematics, for a statement to be true, it must be true all the time. To disprove a statement, you don't need a long essay—you only need to find one single case where the statement fails.

Example:
Statement: "For all prime numbers \(n\), \(n + 1\) is even."
Disproof: Let \(n = 2\) (which is a prime number). Then \(n + 1 = 3\). Since \(3\) is odd, the statement is false.
Counter-example: \(n = 2\).

Common Mistake: Students often try to prove a statement is false by using algebra. While that can work, finding a simple number (a counter-example) is much faster and usually earns full marks immediately!

5. Summary Checklist

Before you finish this chapter, make sure you are comfortable with these skills:

  • Can I use \(\implies\) and \(\iff\) correctly in my working?
  • Do I know the difference between an equation (\(=\)) and an identity (\(\equiv\))?
  • Can I explain why a condition is "necessary" but not "sufficient"?
  • When I see "Disprove the statement...", do I immediately look for a counter-example?
  • When I see "Prove that...", can I try deduction first, and if that's hard, try contradiction?

Don't worry if this seems tricky at first! Proof is a way of thinking, not just a set of formulas. The more you practice looking for "logical holes," the easier it becomes. In the exam, clear communication is key—always explain what you are doing and why your result proves the point!

Note: For more practice on functions or algebra needed for these proofs, see the "Algebra and functions" and "Trigonometry" chapters.