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

**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