Mathematical Proofs - Formulas & Techniques
IB Mathematics Analysis & Approaches
📝 Essential Mathematical Notation
Number Sets:
\(\mathbb{N}\) = Natural numbers (0, 1, 2, 3, ...)
\(\mathbb{Z}\) = Integers (..., -2, -1, 0, 1, 2, ...)
\(\mathbb{Z}^+\) = Positive integers (1, 2, 3, ...)
\(\mathbb{Q}\) = Rational numbers (fractions \(\frac{p}{q}\) where \(p, q \in \mathbb{Z}\), \(q \neq 0\))
\(\mathbb{R}\) = Real numbers
Proof Notation:
LHS = Left-Hand Side
RHS = Right-Hand Side
\(\equiv\) = Identity symbol (true for all values)
\(\implies\) = Implies
\(\iff\) = If and only if (equivalent)
🔢 Representing Integers Algebraically
Any Integer:
\[n \text{ where } n \in \mathbb{Z}\]
Even Integer:
\[2n \text{ where } n \in \mathbb{Z}\]
Examples: ..., -4, -2, 0, 2, 4, 6, ...
Odd Integer:
\[2n + 1 \text{ where } n \in \mathbb{Z}\]
Examples: ..., -3, -1, 1, 3, 5, 7, ...
Consecutive Integers:
\[n, n+1, n+2, \ldots\]
Consecutive Even Integers:
\[2n, 2n+2, 2n+4, \ldots\]
Consecutive Odd Integers:
\[2n+1, 2n+3, 2n+5, \ldots\]
Multiple of k:
\[kn \text{ where } n \in \mathbb{Z} \text{ and } k \text{ is a constant}\]
➡️ Direct Proof (Proof by Deduction)
Definition:
A direct proof starts with known facts, definitions, or axioms and uses logical steps to reach the desired conclusion.
Structure:
1. State what you want to prove
2. Start with known facts or assumptions
3. Use algebra, logic, or theorems step-by-step
4. Arrive at the conclusion
Proving Properties of Integers:
To prove a number is EVEN: Show it can be written as \(2k\) where \(k \in \mathbb{Z}\)
To prove a number is ODD: Show it can be written as \(2k + 1\) where \(k \in \mathbb{Z}\)
To prove divisibility by n: Show it can be written as \(nk\) where \(k \in \mathbb{Z}\)
🔄 Proof by Mathematical Induction
When to Use:
Use mathematical induction to prove statements that are true for all positive integers \(n\), especially for sequences, series, divisibility, and recursive formulas.
Three-Step Structure:
Step 1: Base Case
Prove the statement is true for \(n = 1\) (or the smallest value of \(n\) specified)
Step 2: Inductive Hypothesis
Assume the statement is true for \(n = k\) (where \(k\) is some positive integer)
Step 3: Inductive Step
Using the assumption from Step 2, prove the statement is true for \(n = k + 1\)
Conclusion Statement:
"By the principle of mathematical induction, the statement is true for all \(n \in \mathbb{Z}^+\)"
🔀 Proof by Contradiction
Method:
To prove a statement is true, assume it is false and show this leads to a logical contradiction. Since the assumption leads to impossibility, the original statement must be true.
Structure:
1. Assume the opposite of what you want to prove
2. Use logical reasoning and known facts
3. Arrive at a contradiction (something impossible)
4. Conclude the original statement must be true
Common Applications:
• Proving irrational numbers (e.g., \(\sqrt{2}\) is irrational)
• Proving uniqueness statements
• Proving impossibility statements
• Number theory problems
❌ Disproof by Counterexample
Method:
To disprove a universal statement (a statement claiming something is true for ALL cases), find just ONE specific example where the statement is false.
Structure:
1. Identify the universal claim
2. Find a specific value or case
3. Show the statement fails for this case
4. Conclude the statement is false
Important Note:
ONE counterexample is enough to disprove a universal statement
🔄 Proof by Contraposition
Logical Equivalence:
To prove "If \(P\), then \(Q\)" (written as \(P \implies Q\)), prove the contrapositive: "If not \(Q\), then not \(P\)" (written as \(\neg Q \implies \neg P\))
\[P \implies Q \equiv \neg Q \implies \neg P\]
When to Use:
Use contraposition when the contrapositive statement is easier to prove than the original statement, especially when dealing with "if-then" statements about properties of numbers.
📐 Common Algebraic Identities to Prove
Difference of Squares:
\[a^2 - b^2 \equiv (a-b)(a+b)\]
Perfect Square Trinomials:
\[a^2 + 2ab + b^2 \equiv (a+b)^2\]
\[a^2 - 2ab + b^2 \equiv (a-b)^2\]
Difference of Cubes:
\[a^3 - b^3 \equiv (a-b)(a^2 + ab + b^2)\]
Sum of Cubes:
\[a^3 + b^3 \equiv (a+b)(a^2 - ab + b^2)\]
⭐ Key Formulas to Prove by Induction
Sum of First n Natural Numbers:
\[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\]
Sum of First n Squares:
\[1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\]
Sum of First n Cubes:
\[1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2\]
💡 Proof Writing Tips for IB Exams
✓ State what you're proving at the beginning
✓ Show all steps clearly - don't skip logical connections
✓ Use proper mathematical notation and terminology
✓ Justify each step with reasons or known theorems
✓ Write a clear conclusion linking back to the original statement
✓ Organize your work with clear paragraphs or bullet points
✓ Check edge cases when using algebraic manipulation
✓ State assumptions explicitly (e.g., \(n \in \mathbb{Z}^+\))
🎯 Remember: A proof is a logical argument that convincingly demonstrates why a mathematical statement is true. Practice different proof techniques to build confidence and clarity in your mathematical reasoning.