Venn diagram and Sets

The Venn diagram is more related to the Boolean algebra. The Venn diagram is the pictorial representation of sets that can contain anything from numbers, alphabets, characters, names and anything. These Venn diagrams shows the logical expression of sets.

 

What is Set:

 

A set is a group of collection of numbers, objects, alphabets or names or anything that exist in universe. For example numbers like 2,4,6,8 are individually are integer numbers but as a set they can be considered as a set of even numbers.

 

An overview of older classes:

 

In our old school days, we had studied in our Mathematics subject a chapter of SETs. This chapter is contained in every curriculum of mathematics. A universal set is the one that contains all the elements in the universe. It is represented by a rectangular shape within which many sets are present. Denoted by “U”.

 

Union of Two Sets (OR Logic):

 

The union of two sets is the set that contains all of the elements that are in at least one of the two sets.

 

For example: A = {1, 2, 3, 4} and B = {3, 4, 5, 6}

 

A U B = {1, 2, 3, 4, 5, 6}

 

In digital electronics this union is called as “OR” logic or “OR” gate.

 

Intersection of Two Sets: (AND Logic):

 

The intersection of two sets is the set that contains only the elements that are common in two sets. Other elements not common are excluded.

 

For Example: {1, 2, 3, 4} and B = {3, 4, 5, 6}

 

A  B = {3, 4}

 

In digital electronics, this intersection is called as “AND” logic or “AND” gate

 

Venn diagram:

 

It is the pictorial or diagrammatic representation of sets. For example for above two sets A and B we can draw circles inside the rectangular universal set U.

 

 

 

 

As we can see that the two sets are not overlapping with each other and thus they do not share any common thing. Thus they have nothing in common and shown separately inside the universal set U.

 

We can also see that everything that is outside the circle A or circle B is          or

 

U=UniversalSet=A+

 

 

The A bar sign shows that it is everything except A and same for B

 

CASE-1:

 

Now what happens when there is similarity in the sets? They will intercept and creates Venn diagram like this. We can see that some of the elements of A and B are common and shown in the middle. Partial Overlap.

 

 

 

Boolean Expression = A + B

 

Example: There are total 50 students in the class. 25 are boys (A) and 25 are girls (B). The 5 boys and 5 girls took computer science as major subject. While rest of 20 boys and 20 girls took medical. Hence the intersection of A and B will be total 10 (5 boys and 5 girls)

 

 

 

 

Boolean Expression for the intersecting area will be

 

Boolean Expression = A.B and the region that is not A.B but inside the universal set is

 

 

 

CASE-2:

 

Here we can see that all the elements of set B are present in the elements of set A. But not all the elements of set “A” are present in set B. Set B is a “Subset” of Set A.

 

Boolean Expression = B  A and B = A but A  B

 

Example: Set of positive integers A and set of even numbers B. All the elements of set B are necessarily integers but all the elements of set A are not even numbers set B.

 

 

 

CASE-3:

 

Here we can see that none of the elements of set A and B are common. Hence they are separate. No overlapping.

 

Example: Set of even numbers A and set of odd numbers B. These two sets cannot be identical hence both sets have different elements.

 

Boolean Expression = A ≠ B and B ≠ A

 

 

 

CASE-4:

 

Here we can see that the elements of two sets A and B are exactly identical. Hence they perfectly match each other. Completely Overlap.

 

Example: A = {1, 2, 3, 4} B = {1, 2, 3, 4}. Same / identical sets.

 

Boolean Expression = A =B and B = A

 

 

 

Three Sets:

 

So far we have seen the two sets combination, there can be more sets 3 or 4 or more. Now we will see some of the 3 sets combinations

 

 

 

Here we can see that three sets A, B and C are in intersection with each other. The area that is common all three is the intersection = A  B  C

 

Example: A = {Alex, James, John}

 

B = {Alex, James, Tom}

 

C = {Alex, Tom, John}

 

We can see that in A, B and C the common person is “Alex”. In A and B common person is “James”. In B and C common person is “Tom” and in A and C common is “John”. So all three sets are intersecting with each other.

 

These Venn diagrams are very useful for Karnaugh Map (K-Map) making.

 

Examples-1:

 

Generate the Venn diagram of A’.B

 

Solutions

 

In Boolean algebra, we can denote the bar sign  with A’. Now we have two sets. We will draw A’ first and then B. The notation “.” Is

 

 

 

Now we merge the two figures above. We get the below figure

 

 

 

 

Now we separate the common part between the A’ and B. Remember that the “common” means AND logic. We can see that the intersection of blue and black cross lines forms the desired area for A’B. The region that is outside of A’.B is obviously not A’.B hence we use NOT or bar sign i.e (A’.B)’

 

 

 

Examples-1:

 

Generate B’+A

 

Solution:

 

Here we will draw set B in blue color vertical lines. The set A is in red color vertical lines also. The complimentary of B that is B’ is in black color lines horizontal. Now when we are going to overlap the set B’ with set we will get “hatched” lines For the purpose of OR Function we will check only the hatched lines area. All the area that is hatched with intersection / cross lines and striaght lines in black will be B’+A.

 

 

 

 

Now let’s remove the unwanted region to check the results

 

 

 

This approves the DE Morgan Law from the previous example

 

 

 

 

Remember the principle that .