Tips for Proving Math Theorems

Introduction

• Proving math theorems can be challenging for students, especially in first and second year university math courses.
• This video provides tools to scaffold thinking and offers strategies for constructing proofs.

Step 1: Identify Logical Structure

• Understand the logical structure of the statement to prove.
• Common structure: Conditional Statements (P implies Q).
  • Example: "If X is even, then X squared is even."
• Consider other logical structures like:
  • Biconditional: Both directions must be proven (if P then Q and if Q then P).
  • Existential Claims: Showing existence with formulas (e.g., "There exists an oldest human on Earth").

Step 2: Choose Proof Method

• Different methods depend on the logical structure:
  • Direct Proof: Assume P and manipulate to show Q.
  • Contrapositive: Assume not Q and show not P.
  • Proof by Contradiction: Assume P and not Q, leading to a contradiction.
  • Counterexample: Show that P implies Q is false with one instance where P is true but Q is false.

Step 3: Write Definitions

• Write down definitions for assumptions and conclusions.
• Helps clarify understanding and manipulate terms effectively.

Step 4: Manipulations

• Use mathematical manipulations to connect assumptions to conclusions.
  • Example: For proving "If X is even, then X squared is even":
    • Assume X = 2P (where P is an integer).
    • Manipulate to show X squared = 4P^2 = 2(2P^2), proving that it's even.

Understanding the Statement

• Before proving, understand what the statement is asking.
• Consider the mathematical meanings of assumptions and conclusions.

Visualization and Examples

• Draw geometric representations of the statement to understand connections.
• Use specific examples to gain insight and intuition about the theorem.

Relevant Theorems

• Identify relevant theorems relating to the mathematical objects involved.
• Reading proofs of these theorems can provide ideas for manipulations.

Iterative Process

• Treat proving as an iterative process.
• Experiment with different examples and methods if initial approaches don’t work.

Conclusion

• Encouragement to persist with proving as it can be rewarding and insightful.
• Feedback and questions are welcome for further discussion.