Sufficient or necessary conditions, converse, contrapositive, inverse of an "if P then Q" statement.
Set, empty set, set operations (union, intersection, complement).
Absolute value and intervals.
Ordered pairs, Cartesian product, relation between two sets, reflexive, symmetric, or transitive relation, trichotomy, total order, equivalence relation, equivalence class, set modulo equivalence, well ordered set, complete set.
Functions, injective, surjective, and bijective functions, composition of functions, identity function, inverse function, image and preimage of a set under a function.
Epsilon neighborhoods and deleted epsilon neighborhoods.
Important Proofs
The Theorem of Mathematical Induction.
The Average Theorem.
The Triangle Inequality.
The Archimedean Principle.
The Density Theorem.
De Morgan's laws.
Consequences of the completeness axiom.
Consequences of the field axioms (e.g. -(-a) = a, etc).
There is no rational number whose square is equal to 2.
The First and Second Partition Theorems.
Techniques of Proof
How to prove that a mathematical object (e.g. an additive inverse, a supremum, an inverse function) is unique.