A theorem is a statement that can be shown to be true. A proof is a sequence of statements that demonstrates that a theorem is true. Axioms or postulates are the underlying assumptions about mathematical structures. Proofs may include axioms, the hypotheses of the theorem to be proved, and previously proved theorems. The rules of inference, which are the means used to draw conclusions from other assertions, tie together the steps of a proof.
Fallacies are common forms of incorrect reasoning.
Begging the question or circular reasoning occurs when one or more steps of a proof are based on the truth of the statement being proved. A property of the implication operator which may be confusing is that for the implication to be true, we need only show that p being true implies q is true. By definition, if p is false then the implication is always true, regardless of the truth value of q. This means that a false statement can imply anything whatsoever. Often a direct proof will be more complicated, for example you may have to prove subsidiary results that you later use in one of your implications.