# Introduction; Review of Random Variables

Probability :: Set Theory :: Definition :: Conditional :: Random :: Distribution :: R.V.S.

## Set theory

Probability is intrinsically tied to set theory. We will review some set theory concepts.

We will use to denote complementation of a set with respect to its universe.

- union:
is the set of elements that are in
**
or
**
.

- intersection:
is the set of elements that are in
**
and
**
. We will also denote this as
.

: is an element of the set .

: is a subset of .

: and .

Note that

(where is the universe).

Notation for some special sets:

- set of all real numbers

- set of all integers

- set of all positive integers

- set of all natural numbers, 0,1,2,...,

- set of all tuples of real numbers

- set of complex numbers

That is, if
is a field and
, then
must also be
in
(closed under complementation). If
and
are in
(which we will write as
) then
.

Note: the properties of a field imply that is also closed under finite intersection. (DeMorgan's law: )

What do we mean by countable?

- A set with a finite number in it is countable.
- A set whose elements can be matched one-for-one with is countable (even if it has an infinite number of elements!)

Note: For any collection of sets, there is a -field containing , denoted by . This is called the -field generated by .