Introduction to SETs, Part II

Hello all! Hope you all are safe and fight against ongoing pandemic with positivity.

In my last post Introduction to SETs, Part1, we talked about set theory and focused on explaining different aspects of it like, methods and application of set, types and formal definition of SETs, and many more with examples. As a continuity, my today's post is also on and around set theory to cover the area's that I couldn't in my previous post. Also, like the Part I, this edition also has a video, and I suggest, if someone missed my earlier post and/or video, please check so you can find today's post (Part II) even more relevant.

In this post I will cover,

1. HIERARCHY OF SETS
2. VENN DIAGRAMS

HIERARCHY OF SETS

In Hierarchy of SETs we will learn on how different type of association is setup in case of sets. For say, any two real numbers a and b, this either a=b or a < b or a > b. Here we will consider the sets contained in some other sets and define them with their appropriate designations.

In Hierarchy of sets we will learn,

• Subset
• Proper Subset
• Power set
• Universal set

Subset

Let A and B two sets. Then A is said to be subset of B (or B is super set of A) if every element of A belongs to B and is denoted by A ⊆ B.

“A ⊆ B” read as A is contained in B or A is a subset of B. If we write it as “B ⊇ A” then we read it as B contains A and we call B is a super set of A.

Let take a simple real life example that, A be the set of all red colour pens and B be the set of all pens then obviously all red colour pens are pens first. That is all the members of the set A are members of set B, i.e. A is known as subset of B.

If we see there is at least one element in A (say) which is not in B (say), then set A will not be subset of set B. And it is denoted by A ⊄ B (read as A is not a subset of B).

For example,

i. If A = {1, 3, 6, 9}, B = {1, 2, 3, 5, 6, 8, 9}, then A ⊆ B
ii. If A = {a, b, c}, B = {a, c, d, e}, then A ⊄ B [∵ b ∈ A but b ∉ B ]

Proper Subset

Let take A and B be two sets. Then A is said to be proper subset of B if all the elements of A are in B and B has at least one element other than elements of A and is denoted by A ⊂ B.

For example, if A = {2, 4, 8, 16} and B = {2, 4, 8, 16, 32, 64},
then A ⊂ B [∵ all the elements of A are in B and B has two extra elements, i.e. 32 and 64]

Note:
i. Empty set is subset of every set, i.e. Ø ⊆ A for any set A
ii. Every set is a subset of itself, i.e. A ⊆ A for every set A

Power set

Let A be any set. Then set of all subsets of A is known as power set of A and is denoted by P (A).

Let take an example, if A = {a, b, c, s}, then write P (A).

P (A) = {Ø, {a}, {b}, {c}, {s}, {a, b}, {a, c}, {a, s}, {b, c}, {b, s}, {c, s}, {a, b, c}, {a, b, s}, {a, c, s}, {b, c, s}, {a, b, c, s}}
So, it has 24 = 16 elements.

Note:
i. If A has n elements then total number of subsets of A are 2n.

Universal set

A set U is said to be universal set if all the sets under study are subset of U.

Let define the above formal definition with one real life example. In any particular college there are 9 departments such as Applied Arts and Sciences, Biology, Chemistry and Biochemistry, Computer Science, History, Mathematics, Physics, Political Science, Psychology. If U is the set of all faculties of this college and A1, A2, …, A9 are sets representing the faculties of 9 departments. Then, of course, faculties of all these 9 departments are faculties of this particular college. That is all the members of these 9 departments are present in the set U. Here U plays the role of universal set for the sets A1, A2, …, A9.

Let do solve a simple exercise. If A = {a, c, d, e, d, j}, B = {c, e, f, g, h}, C = {b, i, j}, D = {a, b, j}, then what will be the universal set?

Then universal set U = {a, b, c, d, e, d, g, h, i, j}

VENN DIAGRAMS

As we know that examples play an important role to understand the concepts of theory/definitions. Similarly a diagram speaks more than the words that we may use and they also make the ideas simple and easy to understand even for a fresh reader.

A Swiss mathematician was the first who took the step to represent the sets diagrammatically. Then John Venn a British mathematician, who moving a step ahead simplifying the ideas and made it more user friendly.

Notations used in Venn Diagrams

In Venn diagrams, sets are represented by enclosed areas in a plane as described below:

• Universal set
• Subsets

Universal set

Universal set U is represented by the interior of a rectangle as shown in below figure.

Subsets

Subsets of U are described by the interior of circular discs within the rectangle, representing the universal set U.

When A and B has no common element, i.e. if A ∩ B = Ø

When A and B has some common elements. i.e. A ∩ B ≠ Ø. In general when nothing is mentioned about the common elements of A and B, the below presentation is used.

In the case when A ⊆ B

And when B ⊆ A

So, here I ramp up my both article on Set theory (Part I and Part II).

In next post I will cover,

1. SET OPERATIONS
2. SOME USEFUL AND IMPORTANT LAWS
3. APPLICATION OF SETS

CONCLUSION

Hope you have gained some real knowledge from the article. Please note, this is the second and third video that I have created in a series on SETs. As promised you earlier, through this series, I have covered common yet interesting as well related topic on and around set theory. So watch this space for my next article and video on SETs theory. Till then keep sending me your questions and feedback so I can include those in my upcoming video and/or through my next blog post.