http://www-history.mcs.st-and.ac.uk/history/HistTopics/Beginnings_of_set_theory.html

The history of set theory is rather different from the
history of most other areas of mathematics. For most areas a long process can
usually be traced in which ideas evolve until an ultimate flash of inspiration,
often by a number of mathematicians almost simultaneously, produces a discovery
of major importance.

Set theory however is rather different. It is
the creation of one person, Georg
Cantor. Before we take up the main story of Cantor's
development of the theory, we first examine some early contributions.

http://www.geocities.com/basicmathsets

Welcome to the set tutorial.

To find out how to use this tutorial click
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Below you will see a list of this unit's page headings and the subjects
discussed in each of these pages.

Click on any one of them in order to get started.

##

http://www.cs.odu.edu/~toida/nerzic/content/set/basics.html

Basics of Set

### Subjects to be Learned

- equality of sets
- subset, proper subset
- empty set
- universal set
- power set

### Contents

**Definition (Equality of sets):** Two sets are **equal**
if and only if they have the same elements.

More formally, for any sets *A* and *B*,
*A* = *B* if and only if **
***x* [ *x*
*A*
*x*
*B* ] .

Thus for example **{***1, 2, 3*} = {*3, 2, 1*} , that is the order
of elements does not matter, and **{***1, 2, 3*} = {*3, 2, 1, 1*},
that is duplications do not make any difference for sets.

**Definition (Subset):** A set *A* is a
subset of a set *B* if and only if everything in *A*
is also in *B*.

More formally, for any sets *A* and *B*,
*A* is a **subset** of *B*,
and denoted by *A*
*B*, if and only if **
***x* [ *x*
*A*
*x*
*B* ] .

If *A*
*B*, and *A*
*B*, then *A* is said to be a **proper
subset** of *B* and it is denoted by *A*
*B* .

For example **{***1, 2*}
{*3, 2, 1*} .

Also **{***1, 2*}
{*3, 2, 1*} .

**Definition(Cardinality):** If a set *S* has *n*
distinct elements for some natural number *n*, *n* is
the **cardinality** (size) of *S* and *S*
is a **finite set**. The cardinality of *S*
is denoted by **|***S*|.

For example the cardinality of the set **{***3, 1, 2*} is **3**.

**Definition(Empty set):** A set which has no elements
is called an empty set.

More formally, an **empty set**, denoted
by **,
**is a set that satisfies the following:

**
***x* *x*
,

where
means "is not in" or "is not a member of".

Note that
and {}
are different sets. {}
has one element namely
in it. So {}
is not empty. But
has nothing in it.

**Definition(Universal set):** A set which has all the
elements in the universe of discourse is called a universal set.

More formally, a **universal set**,
denoted by *U* , is a set that satisfies the following:

**
***x* *x*
U .

Three subset relationships involving empty set and universal set are listed
below as theorems without proof. Their proofs are found
elsewhere
.

**Note** that the set *A* in the next four theorems are
arbitrary. So *A* can be an empty set or universal set.

**Theorem 1:** For an arbitrary set *A* *A*
*U* .

**Theorem 2:** For an arbitrary set *A* **
***A* .

**Theorem 3:** For an arbitrary set *A* *A*
*A* .

**Definition(Power set):** The set of all subsets of a set
*A* is called the **power set** of *A*
and denoted by *2*^{A} or ** (***A*)
.

For example for *A* = {*1, 2*}, ** (***A*)
= { ,
{*1*}, {*2*}, {*1, 2*} } .

For *B* = {{*1, 2*}, {{*1*}, 2},
} , **(***B*)
= { ,
{{*1, 2*}}, {{{*1*}, 2}}, {},
{ {*1, 2*}, {{*1*}, 2 }}, { {*1, 2*},
}, { {{*1*}, 2},
}, {{*1, 2*}, {{*1*}, 2},
} } .

Also ()
= {}
and ({})
= {,
{}}
.

**Theorem 4:** For an arbitrary set *A*, the number of
subsets of *A* is *2*^{|A|} .

### Test Your Understanding of Basic Set Concepts