Set Theory

An object in a set is called an **element** of a set or is said to **belong** to the set.If an object x is an element of a set A this is denoted by
$x\in A$

If an object x is not a member of a set A this is denoted by $x\notin A$

A set A is called a **subset** of set B if every element of the set A is an element
of the Set B. If A is a subset of B this is is denoted by
$A\subseteq B$

If set A is not a subset of set B this is denoted by $A\u2288B$

A set A is **equal** to a set B if
$A\subseteq B$
and
$B\subseteq A$

The **intersection** of two sets A and B, denoted by
$A\cap B$
is the set consisting of all elements contained in both A and B.

The **union** of two sets A and B, denoted by
$A\cup B$
is the set consisting of all elements contained in either A or B.

The **set difference**, denoted by
$B-A$
is the set of all elements in the set B that are not in the set A.

The **symmetric difference**, denoted by
$A\u25b3B$
is the set
$\mathrm{(A}-\mathrm{B)}\cup \mathrm{(B}-\mathrm{A)}$

It is easily seen that $A\u25b3B$
= $\mathrm{(A}\cup \mathrm{B)-(A}\cap \mathrm{B)}$

The **empty set** denoted by $\varnothing $ or $\left\{\right\}$

The **universe**, or **universe of discourse** denoted by $U$

Let A be a set. ${A}^{\mathrm{\text{'}}}=U-A$ is the set of all elements not in A.

The **size** or **cardinality** of a finite set A, denoted by
$\left|A\right|$
, is the number of elements in the set.

An infinite set which can be listed so that there is a first element, second< element, third element etc. is called countably infinite.

If it cannot be listed it is said to be uncountable.

Two infinite sets have the same cardinality if there is a one-to-one correspondence between the two sets

We denote this by $\left|A\right|=\left|B\right|$

If there is a one-to-one correspondence between A and a subset of B we denote this by $\left|A\right|\le \left|B\right|$

If $\left|A\right|\le \left|B\right|$ but there is no one-to-one correspondence between A and B we denote this by $\left|A\right|<\left|B\right|$

Let A and B be sets. The **Cartesian product** of A and B, denoted by
$A\times B$
is the set
$\left\{\right(a,b):a\in A\mathrm{and}b\in B\}$

For example let $A=\{a,b\}$ and $B=\{1,2,3\}$ , then $A\times B=\left\{\right(a,1\left)\right(a,2\left)\right(a,3\left)\right(b,1\left)\right(b,2\left)\right(b,3\left)\right\}$

The Cartesian plane $R\times R$ is the set of all ordered pairs of real numbers.

Note that for finite sets $|A\times B|=\left|A\right|\times \left|B\right|$

The **power set** of set A, denoted by
$P\left(A\right)$ is the set of all subsets of A.

For example the power set of
$\left\{\mathrm{a,b,c}\right\}$ is

$\left\{\{a\right\},\left\{b\right\},\{\mathrm{c\},\{a,b\},\{a,c\}\{b,c\},\{a,b,c\},\varnothing \}}$

In the finite case it can be shown that $|P(A\left)\right|={2}^{\mathrm{|A|}}$