What are axioms and theorems?

"Not all proofs are performed in formal logic like this; some use algebra, set theory, or just plain English. But the idea is the same: start with what you know, proceed to derive new knowledge using only legal operations, and end with your conclusion. The things we’re allowed to start with are called axioms (or postulates). An axiom is a presupposition or definition that is given to be true, and so it is legal grounds from which to start. A proof can’t even get off the ground without axioms. For instance, in step 1 of the above proof, we noted that either A or B must be true, and so if B isn’t true, then A must be. But we couldn’t have taken this step without knowing that disjunctive syllogism is a valid form of reasoning. It’s not important to know all the technical names of the rules that I included in parentheses. But it is important to see that we made use of an axiom of reasoning on every step, and that if any of those axioms were incorrect, it could lead to a faulty conclusion. When you create a valid proof, the result is a new bit of knowledge called a theorem which can be used in future proofs. Think of a theorem like a subroutine in programming: a separate bit of code that does a job and can be invoked at will in the course of doing other things. One theorem we learned in chapter 2 was the distributive property of sets; that is, that X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z). This can be proven through the use of Venn diagrams, but once you’ve proven it, it’s accepted to be true, and can be used as a “given” in future proofs."