Important Mathematical Proofs

What is a direct proof?

A direct proof starts with the assumption that the hypothesis is true and uses logical steps to arrive at the conclusion. It involves using definitions, theorems, and axioms to show that if P(x) is true, then Q(x) must also be true.

How does an indirect proof differ from a direct proof?

An indirect proof, such as proof by contrapositive or contradiction, starts by assuming the negation of the conclusion. It then shows that this assumption leads to a contradiction, thereby proving the original statement.

What are common mistakes in mathematical proofs?

Common mistakes include incorrect assumptions, dividing by zero, and misapplying logical steps. Ensuring each step follows logically from the previous one and carefully checking each assumption can help avoid these errors.

A proof is a valid argument that establishes the truth of a mathematical statement. A proof can use the hypothesis of the theorem, if any, axioms assumed to be true, and previously proven theorems. Using these ingredients and rules of inference, the final step of the proof establishes the truth of the statement being proved. But here we will mainly focus on more informal proof.