Introduction to Sets (Also an exercise in Data Visualization)

What are sets?

Definition 1 (from NCERT)

A set is a well-defined collection of objects.

The following points may be noted :
(i) Objects, elements and members of a set are synonymous terms.
(ii) Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
(iii) The elements of a set are represented by small letters a, b, c, x, y, z, etc.

Some examples of sets:
(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) Various kinds of triangles
(v) Prime factors of 210, namely, 2,3,5 and 7
(vi) The solution of the equation: x2 – 5x + 6 = 0, viz, 2 and 3.

We give below a few more examples of sets used particularly in mathematics, viz.
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q + : the set of positive rational numbers, and
R + : the set of positive real numbers.

Definition 2

Sets: grouping of elements and each element in the group is unique.

Two basic types of sets: Overlapping sets and Non-overlapping sets.

Overlapping Sets

If there is one or more element present in both sets, the sets are called to be overlapping.

Example of Overlapping Sets

Set1: First six natural numbers.



Set2: First three natural numbers and first three alphabets.



Now two sets together in a Venn Diagram.





    

        
set1 - set2
        
Intersection
        
set2 - set1
    

    

        

        

        

    

    

        

            Set1 Size:  

            Diff Size: 
        
        
Intersection Size: 
        

            Set2 Size:  

            Diff Size: 
        
    



Example of Non-Overlapping Sets

Non-overlapping sets are better known as Disjoint Sets.

Set1: First six natural numbers.



Set2: First six alphabets.



Now two sets together in a Venn Diagram.





    

        
set1 - set2
        
Intersection
        
set2 - set1
    

    

        

        

        

    

    

        

            Set1 Size:  

            Diff Size: 
        
        
Intersection Size: 
        

            Set2 Size:  

            Diff Size: 
        
    



Nomenclature for sets that are 'Overlapping'

Subset and Superset Relation

If each element of set A is in set B, and also we have some extra elements present in set B which are not in set A:
Then, we call set A a subset and set B a superset.

A superset has more number of elements than a subset.

Set1: Letters from the word 'Rhythm'.



Set2: English Consonants.



Now two sets together in a Venn Diagram.





    

        
set1 - set2
        
Intersection
        
set2 - set1
    

    

        

        

        

    

    

        

            Set1 Size:  

            Diff Size: 
        
        
Intersection Size: 
        

            Set2 Size:  

            Diff Size: 
        
    





Equal Sets

If superset and subset have same number of elements, such that the two sets are fully overlapping each other then we call them 'Equal Sets'. 

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