Types of Mathematical Proofs
Mathematical proofs allow us to verify statements with absolute certainty. A statement is what you're trying to prove, a theorem is a statement proven true, and a conjecture is a statement not yet proven true or false.
Direct proof (or proof by deduction) starts with facts you know and uses logical steps to reach your conclusion. For example, to prove "if x is even, then x² is even," you begin with the definition of even numbers , calculate x² = (2k)² = 4k² = 2(2k²), and conclude that x² must be even because it equals 2 multiplied by another number.
Proof by exhaustion involves breaking your proof into separate cases and proving each one. This works brilliantly when you have a limited number of possibilities to cover. For instance, to prove that "the cube of any natural number is either a multiple of 9, or 1 more/less than a multiple of 9," you would examine three cases: numbers divisible by 3, numbers that are 1 more than a multiple of 3, and numbers that are 1 less than a multiple of 3.
Remember: When choosing a proof method, consider what would be most efficient for the particular statement. Direct proofs work well for straightforward implications, while exhaustion is perfect for statements with a finite number of cases.
