Page 9 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 1.1 Sets

You must have used the word 'set' off and on in your conversations to describe any collection.
In mathematics, the term set is used to describe any well defined collection of objects, that is,
every set should be so described that given any object it should be clear whether the given object
belongs to the set or not.
For instance, the collection N of all natural numbers is well defined, and hence is a set. But the
collection of all rich people is not a set, because there is no way of deciding whether a human
being is rich or not.
If S is a set, an object a in the collection S is called an element of S. This fact is expressed in
symbols as a  S (read as "a is in S" or "a belongs to S"). If a is not in S, we write a  S. For example,
1
3  R the set of real numbers. But   R .
A set with no element in it is called the empty set, and is denoted by the Greek letter  (phi). For
example, the set of all natural numbers less than 1 is .
There are usually two way of describing a non-empty set:

(1) Roster method, and (2) Set builder method.
Roster Method: In this method, we list all the elements of the set within braces. For instance, the
collection of all positive divisors of 48 contains 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 as its elements. So
this set may be written as (1, 2, 3,4, 6, 8, 12, 16, 24, 48).
In this description of a set, the following two conventions are followed:
Convention 1: The order in which the elements of the set are listed is not important.

Convention 2: No element is written more than once, that is, every element must be written
exactly once.

1 1
For example, consider the set S of all integers between 1 and 4 . Obviously, these integers
2 4
are 2, 3 and 4. So we may write S = (2, 3, 4).
We may also write S = (3, 2, 4), but we must not write S = (2, 3, 2, 4). Why? Isn’t this what
Convention 2 says?
The roster method is sometimes used to list the elements of a large set also. In this case we may
not want to list all the elements of the set. We list a few, enough to give an indication of the rest
of the elements. For example, the set of integers lying between 0 and 100 is (0, 1, 2 ,........., 100),
and the set of all integers is Z = (0, +1, !2, ........ }.
Another method that we can use for describing a set is the Set Builder Method.
Set Builder Method: In this method we first try to find a property, which characterises the
elements of the set, that is, a property P which all the elements of the set possess, and which no
other objects possess. Then we describe the set as
{x | x has property P), or as

{x : x has property P).
This is to be read as “the set of all x such that x has property P”. For example, the set of all integers
can also be written as

Z = {x | x is an integer}.

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