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**Set Theory Concepts**

∙ Introduction

∙ Definitions

∙ Universe & Subsets

∙ More on subsets

∙ Operations on sets

∙ Properties of operations

∙ Set graphs & Line graphs

∙ Introduction

∙ Definitions

∙ Universe & Subsets

∙ More on subsets

∙ Operations on sets

∙ Properties of operations

∙ Set graphs & Line graphs

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**:

- each member must be a letter
- 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:

- it must be a soldier
- it must be in the US Army during WW2
- 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.**

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