Venn diagrams

Venn diagrams were first introduced by John Venn to show the connection between different groups of things. Since set is a group of things we use this diagram to explain the relationship between the sets.

             To draw the diagram we first draw a rectangle to denote the universal set.  In that we draw circles or ellipses to denote the subsets.

Example 1:

  Here we are going to see the universal set as set of all alphabets;

          U or  ξ = {set of all alphabets}

Now we want to classify alphabets as vowels and consonants. For that we have to draw two circles which represent vowels and consonants.

Here V stands for vowels and C stands for consonants.

Now we are going to represent the universal set and subsets together.

The above example is for two disjoint(no common elements) sets.

Now let us see an example which have common elements for the subsets.

Example 2 

Draw Venn diagram for the universal set of all natural numbers starting from 1 to 10.

The subsets are A= set of all multiples of 2 < 10

                      B = Set of all multiples of 3< 10

Solution: U = {1,2,3, 4, 5, 6,7, 8, 9, 10}

             A = {2,4,6,8}

             B = [3,6,9}

Here the common element in A and B is 6.

For this we having to draw the two circles in such a way that they they are overlapping each other. In the place which is common to both the circles we have to write the common elements of the two given subsets. There will be some remaining elements which are not in both the subsets. We will write those elements in the area which is outside the circles but inside the rectangle.

Now we are going to see example for proper subset.

Note: We can draw only circles without rectangle if we don't have universal set.

Example 3:

A = { x: x is an even number; 1<x<11}

                  B = {x: x is a multiple of 4:  1<x<9}

Solution: First we write the sets in set builder form.

                  A = {2,4, 6, 8, 10}

                  B = {4, 8}

Here the set B is proper subset of A. To denote this using diagram we have to draw two circles, the smaller one is completely inside the bigger one.

In the next pages we will see the venn diagram for set operations and solving problems using the diagrams.

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