Sets is a well-defined collection of distinct objects. It should be readable and easily understandable. In this chapter, we will discuss the definition of sets, elements of sets, and ways to describe a set.
Summary
A set is always enclosed within the opening and closing of the curly brackets i.e. '{' and '}'. For example: A = {a}.
Elements of sets should always be written in small letters.
No elements in a set can be repeated.
The letter representing a set should always be written in capital form. For example: A = {a} is correct but a = {a} is incorrect way of writing a set.
Introduction to Sets
Set is a well-defined collection of distinct objects.
Well-defined means a set should be understandable. A set should be easy to describe.
Distinct objects means the members in a set should be unique and should not be repeated.
Example of Set:
A = {a,e,i,o,u}
A = {x: x is a set of vowel letters in English alphabet}
Above set is well-defined because we can easily describe the set and is understandable. Anyone can say that it is a collection of vowels in English alphabets. Similarly, it has distinct and unique elements. None of the elements are identical to each other.
Element of Sets
Members of sets are known as elements of that sets. Instead of saying member of sets, we prefer saying 'elements of sets'. This is a term to remember for further study of sets.
In a set A = {a,e,i,o,u}, the members of Set A i.e. 'a', 'e', 'i', 'o', and 'u' are said to be its elements.
Notes
If the elements of a set are alphabets, they should always be written in small letters.
No elements can be repeated within a single set.
Ways of Writing a Set
There are many ways to represent a set. Out of all, only two of the most popular ways of writing a set are explained below:
Listing or Roster or Tabular Method
Set Builder's Method
Listing Method
The listing method is also known as the roster or tabular method. In this method, the elements of a set are listed within two curly brackets (or braces) '{' and '}'.
Example: A = {a,e,i,o,u}
When is Listing method useful?
It can be used to represent sets with less number of elements or a set that has a sequence of elements.
When we have to represent a set with only one element, listing method becomes very much useful because it saves time and space, as well.
Similarly, we can use it to represent a large number of elements but they should follow a sequence. For example: A = {1,2,3,.....,97,98,99}. However, this method is not recommended for educational use. Do not follow this method to represent such large number of elements in an examination.
Set Builder's Method
Set Builder's Method is another way to represent a set in which the elements of a set are represented by variable(s) and value of the variable(s) is/are assigned by using language or mathematical expressions.
Example: A = {x : x is a set of vowel letters of English alphabets}
The set is read as X is such that X is a set of vowel letters of English alphabets. The symbol ':' is read as such that.
How to list elements from a set written using Set Builder's method?
Another example: A = {x: x = 2 + n ; x is natural numbers less than 5}
While writing the elements of this set, we need to loop the given mathematical expression 2 + n for specified number of times to get the distinct elements of the set.
First, we need to know the value of n because that is what value of x depends upon.
Here, n = (1,2,3,4) because it is natural numbers less than 5.
Now,
first element = 2 + n = 2 + 1 = 3
second element = 2 + n = 2 + 2 = 4
third element = 2 + n = 2 + 3 = 5
fourth element = 2 + n = 2 + 4 = 6
So, the elements of the above set are A = {3,4,5,6}.
When is Set Builder's method useful?
Set Builder's method is useful to list any types of sets. However, it becomes really handy when you have to list down a lot of elements, especially infinite number of elements.
For example: A = {x : x is an integer}
It is not possible to list down all the integers within a single set using listing method. So, we can use set builder's method to list down such larger number of elements.