﻿ Well-defined sets & set depiction
Elementary Concepts of Set Theory

# Definition of Sets

Group of things, similar in some clearly recognizable way is called a set.

For instance,

Set of letters from a through e.

Notice that letters f, q, r and z are not members of the set.

Membership in the set, therefore requires two properties:

1. each member must be a letter
2. each member occurs in the position a → e (normal alphabetical order).

A set may be defined by members that must have three properties.

For instance,

Set of soldiers who were in the US Army during WW2 and were <21yrs of age

To be a member in the set, a thing has to have three properties:

1. it must be a soldier
2. it must be in the US Army during WW2
3. it must be < 21yrs old

Therefore,

the number of properties used to define set-membership can be increased to any level we wish.

Why would we want to increase the number of properties?

For a set to be considered in mathematics,

a set must be defined such that it is possible to determine whether or not a given meets the definition.

This is called Well-Defined Set.

Consider a room filled with women.

Then,

the set of women in the room who are under 40yrs old forms a well-defined set.

However,

set of beautiful women in the room is not a well-defined set.

When asked somebody to point out beautiful women, he/she pointing them out might be quite different from another person.

Mathematically, sets that are not well-defined are usually not considered.

## Depiction of a set

The common forms for depicting a set are:

• described
• tabulated (or listing)

For instance,

{a, b, c, d, e} is the tabulated form which may be described as set of letters from a through e.

However, not all sets can be depicted in both described and tabulated forms.

1. If a set contains random collection of items then it may not be described

Example,

collection of a basketball game, a steering wheel, a television etc. may be depicted in tabulated form but may be too inconvenient to be described.

2. If a set contains infinite number of items then it may not be tabulated

Example,

set of citizens of USA is the described form that cannot be tabulated