Direct Proof
Basic terminology
- Definitions, Theorems, Proofs
- Axioms
- Propositions, Lemmas, Corollaries
- Conjectures
Theorems
- Theorem. A theorem is a mathematical statement that is proven to be true.
Theorem: The sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse.
Theorem: The sum of two even integer is an even integer.
Definitions
- Definition. A definition is “an exact unambiguous explanation of the meaning of a mathematical word or phrase.”
Definition: A “right triangle” is a triangle one of whose interior angles is a right angle.
Definition: A positive integer \(n\) is prime if it is greater than one and its only divisors are \(1\) and \(n\).
Definition: A function \(f:\mathbb{R}\to\mathbb{R}\) is continuous at \(x=a\) if \(\lim_{x\to a}f(x)=f(a)\).
- Definitions are sometimes written as “If, Then” but they are really “If and only if” statements.
Proofs
A proof is a logical argument that establishes the truth of a theorem.
A true proof of a mathematical statement is almost never given because of length. In practice a proof describes the key steps that are needed to construct a formal proof. There is a social element in what constitutes a proof which depends on the audience.
Recently some mathematicians have been advocating for computer verified proofs because mistakes do occur in published results.
Lemma, Proposition, Corollary
Lemmas and Propositions are words for “less important” theorems. “Lemma” usually refers to a small theorem that is needed to prove a bigger one. “Proposition” is bigger then “lemma” but smaller than “Theorem.”
Corollary is a word for a theorem that is an immediate consequence of a Theorem.
Lemmas and Propositions preceed theorems; corollaries follow them.
Theorem: Any polynomial function is continuous.
Corollary: Any quadratic function is continuous.