Set

Representation of Set

### Subjects to be Learned

- representation of set
- list up all the members of set
- describe properties for membership for set
- recursive definition

### Contents

A set can be described in a number of different ways.
The simplest is to **list up all of its members**
if that is possible.
For example **{***1, 2, 3*} is the set of three numbers *1*,
*2*, and
*3*. **{** indicates the beginning of the set, and **}** its end.
Every object between them separated by commas is a member of the set.
Thus **{{***1, 2*}, {{*3*}, *2*}, *2*}, {*1* } } is
the set of
the elements **{***1, 2*}, {{*3*}, *2*}
and **{***1*}.

A set can also be described by **listing the
properties**
that its members
must satisfy. For example, **{ ***x*| *1*
*x*
*2*
and *x* is a real number.** }**
represents the set of real numbers between *1* and *2*,
and **{ ***x*| *x* is the square of an integer and
*x*
*100* } represents the set
**{ ***0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100* }.

A third way to describe a set is to give a **procedure
to generate the members
of the set**. The
recursive/inductive definition is an example and it is going to be studied later.

In this representation, first, basic elements of the set are presented. Then a method
is given to generate elements of the set from known elements of the set.
Thirdly a statement is given that excludes undesirable elements
(which may be included in the set otherwise) from the set.
for example the set of natural numbers *N* can be defined
recursively as the set that satisfies the following (1), (2), and (3):

(1) *0*
*N*

(2) For any number *x*
if *x*
*N*,
then
*x + 1*
*N*.

(3) Nothing is in *N* unless it is obtained from (1) and (2).

Following this definition, the set of natural numbers *N* can be obtained as follows:

First by (1), *0* is put into *N*.

Then by (2), since *0* is in *N*, *0 + 1 (= 1)*
is in *N*.

Then by (2) again, *1 + 1 (= 2)* is
in *N*.

Proceeding in this manner all the natural numbers
are put into *N*.

Note that if we don't have (3), *0.5, 1.5, 2.5, ...* can be included in
*N*, which is not what we want as the set of natural numbers.

**
Next -- Equality, Subset, etc. **

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