Welcome to the Chapter on Proof!
Hey there! You've made it to Pure Mathematics 4, and we're tackling one of the most fundamental and rewarding skills in mathematics: Proof.
Why is this important? Proof isn't just about showing your working; it’s about establishing absolute mathematical truth. It's the difference between saying "I think this is true" and "I know this MUST be true, without exception."
This chapter focuses heavily on one powerful, elegant, and often counter-intuitive technique: Proof by Contradiction. Don't worry if this seems tricky at first—we'll break down the logic step-by-step, making sure you grasp the method and can apply it confidently.
What is a Mathematical Proof? (A Quick Refresher)
A proof is a logically sound argument that establishes the truth of a statement (often called a theorem) using previously accepted facts, definitions, or proven theorems.
In simpler terms, you build a sturdy bridge of logic from point A (known fact) to point B (the statement you want to prove).
Key Terminology
- Statement/Proposition: A mathematical claim that is either definitively true or false.
- Theorem: A major statement that has been proven true.
- Axiom: A fundamental statement accepted as true without proof (the absolute starting point).
You may recall Proof by Deduction (or Direct Proof) where you start with a premise and logically follow steps until the conclusion is reached. Proof by Contradiction, however, is an indirect method.
Proof by Contradiction (Reductio ad Absurdum)
This is your main focus in P4. Proof by Contradiction is often summarized by its Latin name: Reductio ad Absurdum, meaning "reduction to absurdity."
The core idea is this: Instead of proving Statement A is true directly, you assume the complete opposite (Not A) is true. If assuming Not A leads you to an impossible mathematical conclusion—a contradiction—then Not A must be false. If Not A is false, then Statement A must be true!
The Analogy of the Broken Phone Charger
You want to prove that your phone's charging port is NOT broken (Statement A).
1. Assume the Opposite: Assume the charging port IS broken (Not A).
2. Test the Consequences: If the port is broken, the phone shouldn't charge even with a known good charger and outlet.
3. The Result/Contradiction: You plug it in, and the charging icon appears immediately (contradicts the assumption that the port is broken).
4. Conclusion: Since the assumption led to a lie, the original statement ("The charging port is NOT broken") must be true.
Step-by-Step Method for Proof by Contradiction
Follow these four critical steps every time you attempt a proof by contradiction:
- Assume the Opposite (The Negation): Start by formally stating that the statement you want to prove is false.
- Logical Deduction: Use this false assumption, combined with definitions and known theorems, to deduce a sequence of logical, algebraic steps.
- Reach a Contradiction: Show that your deduction leads to a statement that is clearly impossible, or contradicts a fundamental mathematical rule or a premise you established in Step 1 (e.g., proving a number is both odd and even, or showing \(1=0\)).
- Conclude: State clearly that because the assumption leads to a contradiction, the initial assumption must be false. Therefore, the original statement must be true.
Classic Example: Proving the Irrationality of \(\sqrt{2}\)
This example is fundamental. You must be able to reproduce this proof.
Statement to Prove (A): \(\sqrt{2}\) is irrational.
Step 1: Assume the Opposite (Not A)
Assume \(\sqrt{2}\) is rational.
If \(\sqrt{2}\) is rational, it can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers, \(b \neq 0\).
CRUCIAL PREMISE: We choose this fraction to be in its simplest form. This means \(a\) and \(b\) are coprime (they share no common factors other than 1).
Step 2: Logical Deduction
We start with \(\sqrt{2} = \frac{a}{b}\).
1. Square both sides:
\(2 = \frac{a^2}{b^2}\)
2. Rearrange:
\(2b^2 = a^2\)
3. Since \(a^2\) is equal to 2 times an integer (\(b^2\)), \(a^2\) must be an even number.
4. A property of integers states: If \(a^2\) is even, then \(a\) must also be even.
5. Since \(a\) is even, we can write \(a = 2k\) for some integer \(k\).
6. Substitute \(a = 2k\) back into the equation \(2b^2 = a^2\):
\(2b^2 = (2k)^2\)
\(2b^2 = 4k^2\)
7. Divide by 2:
\(b^2 = 2k^2\)
8. Since \(b^2\) is equal to 2 times an integer (\(k^2\)), \(b^2\) must be even. Therefore, \(b\) must also be even.
Step 3: Reach a Contradiction
From the deduction, we concluded that:
- \(a\) is even.
- \(b\) is even.
This means \(a\) and \(b\) share a common factor of 2.
BUT, in Step 1, we defined \(a\) and \(b\) as having no common factors (coprime).
This is the CONTRADICTION (It contradicts our initial premise in Step 1).
Step 4: Conclude
Since the assumption that \(\sqrt{2}\) is rational leads to a logical impossibility, the assumption must be false. Hence, \(\sqrt{2}\) must be irrational.
Common Mistakes to Avoid in Contradiction Proofs
- Skipping the Premise: You MUST state that \(a\) and \(b\) are coprime (in their simplest form) at the beginning of the irrationality proof. If you miss this, you lose the basis for the contradiction.
- Circular Logic: Ensure your contradiction is genuine—it must break a known fact or your own stated premises, not just be confusing algebra.
- Ignoring Negation: When asked to prove "The statement is TRUE," you must start by assuming "The statement is FALSE."
Did you know? Proving the irrationality of \(\sqrt{2}\) is one of the earliest known proofs, often attributed to the Pythagoreans. It was a revolutionary idea that shocked the ancient world because they believed all numbers could be expressed as ratios!
Disproof: Showing a Statement is False
Proof by Contradiction establishes absolute truth. However, if a statement is generally false, we use a simpler tool: Disproof by Counterexample.
Disproof by Counterexample
If a statement claims something is true for all values (universal statement), you only need to find one single instance where the statement fails to prove that the entire statement is false.
That single failure is the counterexample.
Example: Disprove the statement: 'For all prime numbers \(p\), \(p+2\) is also a prime number.'
We check cases:
- If \(p=3\), \(p+2=5\). (True)
- If \(p=5\), \(p+2=7\). (True)
- If \(p=7\), \(p+2=9\). (9 is not prime).
Counterexample: The case where \(p=7\) produces a non-prime number (9). Since the statement failed for \(p=7\), the original claim ("For all prime numbers \(p\), \(p+2\) is also a prime number") is false.
Memory Aid: Proof vs. Disproof
Proof requires logical steps that cover All possible scenarios.
Disproof requires finding just One failure (the Counterexample).
Chapter Summary & Final Encouragement
Proof is perhaps the most intellectually demanding part of Pure Maths, but it's also the most rewarding. It requires attention to detail and absolute rigor. Don't worry if it takes time to master the structure.
Quick Review: Proof by Contradiction Strategy
1. Assume the Opposite: What if the statement was false? Establish all initial premises (like coprime integers).
2. Deduce: Follow the logical steps rigorously until you can't go further.
3. Contradiction: Hit an impossibility (like an even number must equal an odd number, or contradicting your premise from Step 1).
4. Conclude: Since the assumption led to an absurdity, the original statement must be TRUE.
Keep practicing the standard proofs, focusing especially on the irrationality proof structure. You've got this!
The key to success is practicing the structure, not just memorizing the steps.