Introduction

Prerequisites for the course:

Differential Calculus
Integral Calculus

Set Theory

Definitions:[Set]
A set is a collection (possibly empty) of distinct objects. Note that the order that the elements of the set does not matter.

Definition:[Empty Set]
The empty set, or null set, is a set which does not contain any elements. We denote the empty set $\emptyset$ .

Definition:
We say that $E$ is a subset of $F$ if all elements of $E$ are also elements of $F$ . We denote a subset by $E \subset F$ .

Definition:[Super Set]
If $A \subset B$ we then say that $B$ is a super set of $A$ , that is $B$ is a set which contains $A$ . We denote a super set as $B \supset A$ .

Example:
If $A = \set{1,2,3}$ and $B = \set{0,1,2,3,4}$ then we have that
\begin{itemize}
\item $A \subset B$
\item $B \supset A$
\end{itemize}
which one we use depends on the application we are working with.

Definition:[Power Set]
For a set $X$ , the power set of $X$ , denoted $\mathcal{P}(X)$ , is the collection of all subsets of $X$ .

Definition:[Set Equality]
We say that $E$ and $F$ are equal if $E \subset F$ and $F \subset E$ . That is both $E$ and $F$ have all of the same elements.

Definition:[Intersection]
A set is the intersection of two sets if it contains all elements that are shared between both sets. That is

$\begin{align*} E \cap F &= \set{x \in X | x \in E \text{ and } x \in F} \end{align*}$

Example:
If $A = \set{1,2,3,4}, B = \set{3,4,5,6}$ then $A \cap B = \set{3, 4}.$

Example:
If $A = [1,5], B = (2,6]$ then $A \cap B = (2,5].$

Definition: A set is said to be the union of two sets if it contains elements that occur in either of the sets (or both). We denote it by $\cup$ and it is defined to be

$\begin{align*} E \cup F &= \set{x \in X | x \in E \text{ or } x \in F} \end{align*}$

Definition:
A set is called the complement of an set, $E$ , if it contains all of the points that are not in $E$ and none of the points that are in $E$ . We denote the complement by $E^c$ and define it to be

$\begin{align*} E^c &= \set{x \in X | x \not\in E} \end{align*}$

Note that when we take the complement of a set it is in regards to another superset.

Definition:{Difference} – An event is called the difference of two events $E$ and $F$ if it occurs whenever $E$ occurs but $F$ does not. We denote the set difference by $E - F.$ We have that $E^c = S - E$ and $E - F = E \cap F^c$ . In mathematical terms

$E - F = \set{x \in S | x \in E \text{ and } x \notin F}$

Definition:
We say that two sets, $E, F$ , are mutually exclusive if $E \cap F = \emptyset$ .

Example:
If $E$ is the set of all odd numbers and $F$ is the set of all even numbers then $E \cap F = \emptyset$ and so $E$ and $F$ are mutually exclusive.

Set Operation Properties

(1) $\begin{align*} (E^c)^c &= E \\ E \cup E^c &= S \\ E \cap E^c &= \emptyset \end{align*}$

Definition:{Commutative Laws:}

(2) $\begin{align*} E \cup F &= F \cup E \\ E \cap F &= F \cap E \end{align*}$

Definition:{Associative Laws:}

(3) $\begin{align*} E \cup (F \cup G) &= (E \cup F) \cup G \\ E \cap (F \cap G) &= (E \cap F) \cap G \end{align*}$

Definition:{Distributive Laws:}

(4) $\begin{align*} (E \cap F) \cup G &= (E \cup G ) \cap (F \cup G) \\ (E \cup F) \cap G &= (E \cap G ) \cup (F \cap G) \end{align*}$

Definition:{De Morgan’s First Law:}

(5) $\begin{align*} (E \cup F)^c = E^c \cap F^c \end{align*}$

Definition:{De Morgan’s Second Law:}

(6) $\begin{align*} (E \cap F)^c = E^c \cup F^c \end{align*}$