## 1. Definitions and Axioms

Primitive notion: A set is a collection of objects. The objects in the collection are called elements of the set. constructed using the following definitions and axioms.

This is not a definition because we do not define "collection" or "object". Not all collections are sets. Sets will be collections which can be constructed from the following definitions and axioms.

Definition 1.1: Let A and B be two sets. If every element of A is also an element of B then we say that A is a subset of B, denoted by.

Axiom 1: If A is a set and B is a subset of A, then B is also a set.

Definition 1.2: Let A and B be two sets. A = B if and only if

Theorem 1.1: If A is a set then

Definition 1.3: Let A and B be two sets. The union of A and B, denoted by

is the set of things that are either elements of A or elements of B.

Axiom 2: If A and B are sets then so is their union.

Theorem 1.2: Let A and B be two sets, then

Definition 1.4: Let A and B be two sets. The intersection of A and B denoted by

is the set of things which are elements of both A and B.

Theorem 1.3: Let A and B be two sets.

Theorem 1.4: If A and B are two sets, the so is their intersection.

Definition 1.5: Let A and B be two sets. The difference of A and B is the set of things which are in A but not in B. The difference of A and B is denoted by A - B.

Definition 1.6: Let U be a universal set and let A be a subset of U. Then the complement of A with respect to U is U - A. The notation for the complement of A will be A'

Theorem 1.5: If A and B are sets then so is A - B.

Theorem 1.6: The complement of a set is a set.

Definition 1.7: Let S be a set. The set of all subsets of S is called the power set of S and is denoted by P(S).

Axiom 3: If S is a set then so is its power set.

Definition 1.8: Let a and b be two elements. The ordered pair of a and b denoted by (a, b ) is defined to be

(a, b ) = {{ a }, { a, b }}

Theorem 1.7: (a, b ) = (c, d ) if and only if a = c and b = d.

Definition 1.9: Let A and B be two sets. The Cartesian product of A and B, denoted by A x B is the set of all ordered pairs whose first coordinate comes from A and whose second coordinate comes from B.

Theorem 1.8: The Cartesian product of two sets is a set.

Convention: Order of operations: If you have a procedure where new sets are being produced from existing sets by taking unions, intersections, differences, and Cartesian products, then we will agree that unless there are parentheses, we will take

• Cartesian products first,
• intersections and differences next and
• unions last.

If there are parentheses they tell us to do the operations in parentheses in the order specified above before doing anything else.

Theorem 1.9: Let A be a set. Then

Theorem 1.10: Let A be a set. Then

2. Set Operations