What is a Mathematical Proof?

To prove a mathematical statement, follow these steps:

Understand the Statement Clearly

Carefully read the given problem. Identify the hypothesis (given conditions) and the conclusion (what needs to be proven).

Gather Known Facts

Recall relevant definitions, axioms, postulates, and previously proven theorems that might help in the proof. Identify logical properties and relationships between mathematical objects in the problem.

Choose a Proof Technique

There are different methods of proving mathematical statements. Chose a suitable one.

Direct Proof

Start from known facts and logically deduce the conclusion. It is commonly used for algebraic and number theory proofs.

Example: Proving that the sum of two even numbers is even.

Indirect Proof (Contradiction)

Assume that the statement is false and show that this assumption leads to a contradiction.

Example: Proving that 2is irrational.

Contrapositive Proof

Instead of proving “If P, then Q,” prove the equivalent contrapositive statement: “If NOT Q, then NOT P.”

Divide the problem into different possible cases and prove the statement for each case separately.

Example: Proving properties of absolute values.

Proof by Induction (For statements involving natural numbers)

Base case: Verify the statement for the first value (usually n=1).

Inductive step: Assume the statement is true for n = k and prove it for n = k+1.

Proof by Construction

Directly construct an example that satisfies the given conditions.

Proof by Counterexample (For Disproving Statements)

Find one specific example that contradicts the given statement.

Example: Disproving “All prime numbers are odd” (counterexample: 2).