Notes on Proof in Mathematics

Proving Mathematical Theorems and Statements

Definition of Proof

• Proof: A proof is a logical argument establishing the truth of a mathematical statement. Proofs are essential in mathematics; they provide a foundation for theorems and help in validating claims.

Types of Proofs

Direct Proof

• A direct proof demonstrates the truth of a statement by straightforward logical deductions from accepted premises or previously established results.

Indirect Proof

• Indirect Proof: Also known as proof by contradiction, this method assumes the negation of the statement to be proven and derives a contradiction.

Contrapositive Proof

• A proof that involves proving the contrapositive of a statement, which is logically equivalent to the original statement.

Mathematical Induction

• A technique used to prove statements or formulas that are typically stated for all natural numbers. The process involves two steps:
  • Base Case: Show that the statement holds for the initial value (usually n=1).
  • Inductive Step: Assume the statement holds for n = k, then show it also holds for n = k + 1.

Examples of Proof

Example of Direct Proof

• Theorem: For any two odd integers a and b, their sum is even.
  • Proof: Let a = 2m + 1 and b = 2n + 1, where m and n are integers.
  • Calculate a + b:
    $a + b = (2m + 1) + (2n + 1) = 2m + 2n + 2 = 2(m + n + 1)$
  • Since (m + n + 1) is an integer, a + b is even.

Example of Indirect Proof

• Theorem: The square of an irrational number is irrational.
  • Assumption: Assume the contrary; thus, the square of an irrational number is rational. Let ( x ) be an irrational number such that ( x^2 ) is rational. Then ( x^2 = \frac{p}{q} ), where p and q are integers and q ≠ 0. Using this, derive a contradiction.

Importance of Proofs

• Provides validation and assurance of the truth.
• Facilitates the understanding of deeper mathematical concepts and relationships.
• Builds logical thinking, which is essential in mathematics and other fields.