Introduction to SETs, Part III

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

In my previous posts Introduction to SETs, Part1 and Introduction to SETs, Part II, 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 III) even more relevant.

In this post I will cover,

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

SET OPERATIONS

In my previous post Introduction to SETs, Part I and Introduction to SETs, PartII, I have talked about the definition and types of sets, Hierarchy of SETs and Venn Diagrams. Now we will learn about some commonly used operations on sets. Those are,

• Union
• Intersection
• Complement of a Set
• Difference of Two Sets
• Symmetric Difference of Two Sets

Union

Let A and B be two sets then union of A and B is denoted by A ∪ B and is defined as
A ∪ B = {x: either x ∈ A or x ∈ B}

i.e. A ∪ B contains all the elements of A as well as of B.

                                                        Shaded region denoted A ∪ B

Let A be a set containing the footballers in Europe getting salary (in €) between 5 million to 10 million per year and another set B containing the footballers in Europe getting salary (in €) between 1 million to 7 million per year. For this example, if we are interested in finding those footballers who are getting the salary within the range 5- 10 or 1-7 then such footballers will be those having salary between 1 million and 10 million. The set of such footballers is nothing but the union of two sets A and B.

More Examples,

i. If A = {5}, B = {d, e, g}, then A ∪ B = {5, d, e, g}
ii. If A = {3, 5, 6}, B = {1, 2, 3, 5}, C = {1, 4, 6}, then A ∪ B ∪ C = {1, 2, 3, 4, 5, 6}

Intersection

In intersection, let A and B two sets then intersection of A and B is denoted by A ∩ B and is defined as

A ∩ B = {x: x ∈ A and x ∈ B}

i.e. A ∩ B contains common elements of A and B.

                                                        Shaded region denotes A ∩ B

From the above example on footballers in Europe getting salary, if we are interested in finding those footballers who are getting the salary within the common range then such persons will be those having salary between 5 million to 7 million per year. The set of such footballers is nothing but the intersection of two sets A and B.

If A ∩ B = Ø then, the that two sets A and B are disjoint.

Examples,

i. If A = {}, B = {1, 3, 9}, then A ∩ B = Ø
ii. If A = {45, 52, 31}, B = {5, 7, 9}, then A ∩ B = {} = Ø = empty set, as there is no common element in the two sets.
iii. If A = {b, d, f, h}, B = {a, b, c, e ,g}, then A ∩ B = {b}

Complement of a Set

Let U be the universal set, then complement of a set A (where A ⊆ U) is denoted by Ac or A or A’ and is defined as A’ = {x ∈ U: x ∉ A}

That is A’ contains those elements of U which are not in A, i.e. A’ contain all elements of U other than A.

                                            Shaded region denotes the complement of the set A i.e. A’

Suppose we have a set of persons of a locality having voting right. Then set of those persons of the locality who do not have voting right is its complement, if the set of all persons of that locality is considered as a universal set.

Example,

i. If U = {a, d, e, f, g, h, i, j, l, m, o, r, u} and A = {d, f, g, h, j, l, m, r}, then A’ = {a, e, i, o, u}, i.e. those elements of U which are not in A.

Difference of Two Sets

Let A and B be two sets then difference of A and B is denoted by A – B and is defined as
A - B = {x: x ∈ A but x∉ B}

                                             Shaded region denotes A – B

For examples,

i. If A = {1, 2, 3}, B = {1, 2, 3, 4, 5}, then A-B = { }
ii. If A = {a, b, c, d, e}, B = {d, e, f, g}, then A – B = {a, b, c}

Symmetric Difference of Two Sets

Let A and B be two sets, then symmetric difference of A and B is denoted by A △ B and is defined as
A △ B = (A - B) ∪ (B - A)

                                              Shaded region denotes A △ B

For example, if A = {a, b, c, d}, B = {c, d, e, f, g}
Then, A – B = {a, b} and B – A = {e, f, g}
\ A △ B = (A - B) ∪ (B - A) = {a, b} ∪ {e, f, g} = {a, b, e, f, g}

CONCLUSION

Hope you have gained some real knowledge from the article. Please note, this is the first, second, third and fourth 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 fundamental of mathematics. 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.