Deductive proof, equality and identity

Introduction to Proof

In most of your mathematics journey, you have been "solving" problems to find a specific value for \(x\) or \(y\). However, in this chapter, we move into the heart of pure mathematics: Proof. Instead of finding what the answer is, we are focusing on why a statement is true for every possible number. Proof is about building a logical bridge from what we know to be true to a new conclusion that cannot be disputed.

Did you know? In mathematics, a "conjecture" is a statement that people think is true but hasn't been proven yet. Once it is proven using the methods we will discuss, it becomes a "Theorem."

1. Equality vs. Identity

Before we start proving things, we need to understand the difference between an equation and an identity. This is the foundation of the Number and Algebra section.

Equations (Equality)

An equation uses the symbol \(=\). it is a statement that is true only for some values of the variable.
Example: \(2x + 1 = 5\). This is only true when \(x = 2\). If \(x = 3\), the equality breaks.

Identities

An identity uses the symbol \(\equiv\). It is a statement that is true for all values of the variable.
Example: \(2(x + 3) \equiv 2x + 6\). No matter what number you plug into \(x\), the left side will always equal the right side.

Key Takeaway: In a proof, we are often trying to show that an expression is an identity—that it works for every possible number \(n\) or \(x\).

2. Simple Deductive Proof (SL & HL)

A deductive proof starts with known facts (like algebraic rules) and moves step-by-step to a conclusion. For the IB exam, the most important technique is the LHS to RHS method.

How to lay out a proof:

1. Start with the LHS (Left Hand Side) of the expression.
2. Use algebraic manipulation (expanding, factoring, simplifying).
3. Show that it eventually looks exactly like the RHS (Right Hand Side).

Important Rule: Never "move" terms across the equals sign like you do when solving equations. Keep the sides separate until they match!

Example: Prove that \((n + 1)^2 - (n - 1)^2 = 4n\)
LHS: \((n + 1)^2 - (n - 1)^2\)
\(\implies (n^2 + 2n + 1) - (n^2 - 2n + 1)\)
\(\implies n^2 + 2n + 1 - n^2 + 2n - 1\)
\(\implies 4n\)
RHS: \(4n\)
Since LHS = RHS, the proof is complete.

Quick Review: To prove something is true, you must show it works for the general case (using letters like \(n\)), not just by testing one or two numbers.

3. Proof by Counterexample (AHL Only)

Sometimes you are asked to show that a statement is false. To do this, you don't need a long logical chain; you only need one example where the statement fails. This is called a counterexample.

Example: "All prime numbers are odd."
To prove this is false, we just need one counterexample. The number \(2\) is a prime number, but it is even. Therefore, the statement is false.

Common Mistake: Students often try to find a "general" reason why something is false. You don't need to! Just find one single number that breaks the rule.

4. Proof by Contradiction (AHL Only)

This is a "backdoor" way of proving something. Instead of proving it is true directly, we assume it is false and show that this assumption leads to something impossible (a contradiction).

The Steps:

1. Assume the opposite of what you want to prove.
2. Follow the logic until you hit a "clash" (e.g., you find that a number must be both even and odd).
3. Conclude that because the assumption led to nonsense, the original statement must have been true.

Analogy: If I want to prove I am not in Paris, I could assume I am in Paris. But if I am in Paris, I should be able to see the Eiffel Tower. I look around and see the Sydney Opera House. This is a contradiction, so my assumption (being in Paris) must be wrong.

5. Proof by Mathematical Induction (AHL Only)

Induction is used to prove that a statement is true for all positive integers \(n \in \mathbb{Z}^+\). Think of it like a row of falling dominoes.

The "Domino" Steps:

Step 1: The Base Case. Show the statement is true for the first domino (\(n = 1\)).
Step 2: The Assumption. Assume the statement is true for a random domino in the middle (\(n = k\)).
Step 3: The Inductive Step. Use the assumption from Step 2 to prove the next domino (\(n = k + 1\)) must also fall.
Step 4: Conclusion. Write the formal concluding sentence.

Standard Conclusion Text:
"Since the statement is true for \(n = 1\), and if true for \(n = k\) it is shown to be true for \(n = k + 1\), then by the principle of mathematical induction, the statement is true for all \(n \in \mathbb{Z}^+\)."

Don't worry if this seems tricky at first! Induction is a very specific "recipe." Once you learn the steps for sums, products, or divisibility, the algebra starts to look very similar every time.

Summary Checklist for the Exam

1. SL/HL: Do I know the difference between \(=\) and \(\equiv\)?
2. SL/HL: Can I layout a proof by starting at the LHS and working to the RHS?
3. AHL: Can I find a counterexample to disprove a false claim?
4. AHL: Do I remember to assume the opposite at the start of a contradiction proof?
5. AHL: Have I memorized the formal conclusion for Induction?

Note: For more practice on sequences which often appear in induction proofs, see the "Arithmetic sequences and series" or "Geometric sequences and series" chapters.