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What is a Theorem? What is a Proof?

Discrete Mathematics Basic Direct Proofs What is a Theorem? What is a Proof?

Throughout your mathematical journey, you’ve come across several different theorems, such as “The Fundamental Theorem of Arithmetic,” “The Pythagorean Theorem,” “The Fundamental Theorem of Calculus, or Algebra,” or “The Quotient Remainder Theorem.” So, what’s a theorem?

Definitions

A Theorem is a nontrivial (i.e. “not obvious”) mathematical statement that has been proven to be true by inferentially reasoning from definitions, a set of commonly accepted simple assumptions called axioms, or from other theorems. Such an argument is called a proof of a mathematical statement.

A statement that has NOT been proven true is called a claim, or a conjecture.

A lemma is a “helping theorem;” i.e. a “smaller” theorem that helps to prove a bigger, more important theorem.

A corollary is a theorem that follows easily or immediately as a result of another theorem.

Helpful Note

Loosely speaking, a proof is a list of true statements that “explain why” a conjecture is true.

It is worth noting that, because theorems are proved by deductively reasoning from a set of core assumptions, there is no “wiggle room” in terms of how true they are. If a statement has been (correctly) proven true, then there is no exception to the mathematical statement given. For instance, one can show that every number divisible by $4$ is an even number. Because this statement is true and can be proved deductively, we are guaranteed that you will NEVER find a number divisible by $4$ that is NOT even!

This strongly contrasts with any other field of study where statements are made and “proved” inductively. All reasoning performed in scientific fields establish “reasonable expectation” of a particular outcome, not “absolute undeniable certainty, without exception.”

In the lessons that follow, we will talk about proving statements directly.

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