Introduction to SETs, Part1

Welcome to my blog. In today’s post, we will talk about set theory. Well the theory is already cleared to many but this edition of my article is dedicated to those who wants to dig deep. I have created a video and embedded it here so you can understand better the concepts like, what is set, methods of set, application of sets and many more with simple examples. In addition to that, I also covered, types and hierarchy of sets, operations of sets, and Venn diagram which is pictorial representation of sets.

I have divided SET theory into three parts,

1. Part 1 (this post)
2. Part 2
3. Part 3

Let’s begin –

In this post I will cover,

1. Introduction to SETs
2. Methods of representing a SET
3. Types of SETs

Formal definition of SETs

A well-defined collection of distinct objects is called a set. A set is generally denoted by capital letters, such as A, B, C… etc. And the objects which belong to the set are known as elements or member of the set and are denoted by small letters x, y, z… etc.

If ‘a’ is an element of a set B then we write
a ∈ B (read it as ‘a’ belongs to B)

If ‘a’ is not an element of a set B then we write
a ∉ B (read it as ‘a’ doesn’t belong to B)

A set remains the same if some or all of its elements are repeated or rearranged. For example, if a set contains the elements 0, 1, -1 and another set contains the elements -1, -1, 1, 0, 0, 1, 1, 0; then these two sets are nothing but represent the same set having three elements 0, 1, -1. As we see repetition of elements in a set is not allowed, so once a set is defined, automatically number of elements contained by it has also become fixed.

Let see some basics examples

1. Collection of letters of the word “SETS” – It is a set and elements of this set are S, E, T

2. Collection of good footballers from Brazil – This collection does not form a set, because a given player may be good according to some person but the same player may not be good according to some other person.

Methods of representing a SET

A set is generally represented by two methods,
1. Roster method / Tabular form / Listing method
2. Set-Builder method / Property method / Rule method

1. Roster method

In this method each and every element is listed and put, separating by commas, in curly brackets.

For example,
i. If A is the set of vowels of English alphabets, then A = {a, e, i, o, u}
ii. If N is a set of Natural numbers, then N = {1, 2, 3, …}

Now, we will write the following set using by roster method,
C = {x: x2 – x – 30 = 0, x ∈ N}

C = {x: x2 – x – 30 = 0, x ∈ N}
= {x: x2 – 6x + 5x – 30 = 0, x ∈ N}
= {x: x(x - 6) + 5(x - 6) = 0, x ∈ N}
= {x: (x - 6) (x + 5) = 0, x ∈ N}
= {x: x = 6, -5, x ∈ N}
= {6} [∵ -5 ∉ N]

2. Set-Builder method

In this method we consider one or more properties that are exclusive to the elements of a set so that no other elements can be member of the set.

For example,

Let A = {x: x is a vowel of English alphabets}, then elements of A are a, e, i, o, u and having exclusive property of being a vowel no other alphabets can be considered as an element of set A.

(here “:” read as such that )

In some situations we have advantages of Set-Builder methods, like sometimes we cannot list the elements of the set or even if we can list them, it may not be practical or feasible to do so.

For example, consider the set {x: x is a real number and 1 < x < 9}. Because number of elements in this in this set is uncountable, so this set cannot be described by roster method.

Now, we will write the following set using by set-builder method,
E = {2, 4, 6, 8, 10, 12, …}

Here we see that elements in this set are multiple of 2 so in set-builder form it can be written as,
E = {x: x is a multiple of 2, x ∈ N}

Types of SETs

Different names given to a set on the bases of the number of elements contained by the set are,

1. Null set
2. Singleton set
3. Finite set
4. Cardinal number of a finite set
5. Infinite set
6. Equivalent sets
7. Equal sets

1. Null set

A set is said to be Null set (or empty or void), if it has no element in it. Null set is denoted by ∅ or {}

For example, consider the collection of those sons having their ages more than their respective fathers. Obviously this is not possible. So this is called Null set.

2. Singleton set

A set is said to be singleton set if it contains only one element. Consider the collection of mothers of a baby. Obviously a baby has only one mother. This type of collection having a single element is known as singleton set.

For example, A = {x: x is an even prime number} is a singleton set as there is only one even prime number, i.e. 2

3. Finite set

A set is said to be finite set if it is an empty set or it has a finite number of elements.

For example, A = {3, 7, 2, 9, 8} is a finite set because it contains 5 elements, i.e. finite number of elements.

4. Cardinal number of a finite set

The number of elements in a finite set says A is called its cardinality and is denoted by n(A).

For example, If A = {3, 7, 2, 9, 8} then n(A) = 5. i.e. cardinality of A is 5

5. Infinite set

The not finite set is said to be Infinite set. Infinite sets are either countable or uncountable.

For example, A = {3, 7, 2, 9, 8, …} is an infinite set.

6. Equivalent sets

Two finite sets A and B (say) are said to be equivalent if number of elements in both the sets are equal in numbers, i.e. n(A) = n(B) and we denote it by A ~ B (read as A is equivalent to B).

For example, if A = {x, y, z} and B = {8, 9, 10}, then

A ~ B [∵ n(A) = n(B) = 3]

7. Equal sets

Two sets A and B are said to be equal if every element of A is in B and every element of B is in A and is written as A = B.

For example, if A = {x, y, z, g} and B = {z, g, y, x}, then
A = B

• Here order of elements does not matter.

• If two sets A and B are not equal then we write A ≠ B.

In my next post on SET theory I will cover,

1. HIERARCHY OF SETS
2. VENN DIAGRAMS

Conclusion

Hope you have enjoyed the video and reading the article. Well, this is the first video that I have created in a series on SETs. Through this series, I will cover many more interesting as well related topic like, hierarchy of sets, set operators, some useful and important laws applying on sets, and more on Venn diagram. So stay tuned for my next article. Till then keep sending me your queries if you have anything so I can cover those in my upcoming video and/or through my next blog post.