Elementary Concepts of Set Theory

Definition of Sets

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


For instance,

Set of letters from a through e.

f, q, r, z 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).



WW2 soldiers < 21yrs 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



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.


Women in a room

Consider a room filled with women.



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




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.

ill-defined set


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


Depiction of a set


The common forms for depicting a set are:


For instance,


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



random collection 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



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



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