Describing sets

In this post we will learn different methods to describe the sets with solved examples.

This chapter is important because here you will learn how to write sets on your own.

Before reading the chapter, make sure you have cleared your basics on sets which have been discussed in previous chapter.

How to describe a set?

Before we learn the methods to represent the sets, let us do the basic review of the sets.

What are sets?

Sets are the collection of objects in a structured manner.

These objects are called elements or members of the sets. The elements of the sets are written inside the closed curved brackets.

Given below are some examples of sets:

(a) X = { 6, 31, 42, 50}

Here the set name is X and it contains four elements, namely, 6, 31, 42 & 50.

(b) Y = { -3, 7, -32}

Here the set name is Y and it contains three elements namely, -3, 7 & -32.

Methods to describe a set

Here we will learn three methods to describe a set

(a) Statement form
(b) Roster or Tabular form
(c) Set Builder form

Let us discuss each of the methods step by step.

Statement Form of set representation

In Statement form, the sets & its elements are defined clearly in words.

Note that the sentence will be enclosed in the curly bracket.

For your understanding, just think that instead of writing each element in the curly bracket, we are describing the elements in form of words.

Examples of Statement form of sets

(i) X = { Natural number multiple of 7 }

Here the set name is X.
It contains elements which are natural number which is multiple of 7.

Note:
Observe that in Statement form, the individual elements are not listed. It just give us a hint about the kind of element present in the set through a statement.

(ii) Y = { Numbers greater than 50 & less than 60 }

The set name is Y.
It contain numbers which are greater than 50 but less than 60.

(iii) Z = { USA Rugby players between age 20 to 25 }

Here the set name is Z.
It contains Rugby players name from age 20 to 25

Roster or Tabular form of Set Representation

In Roster form, we individually list the elements inside the curved brackets separated by commas.

Examples:

(i) Set X in Statement form
X = { Vowels of English alphabets }

Set X in Roster form
X = { a, e, i, o, u }

Note:
Observe that in roster form, we have individually listed all the elements inside the curved bracket.

(ii) Set Y in Statement form
Y = { Number greater that 30 but less than 35 }

Set Y in Roster Form
Y = { 31, 32, 33, 34 }

(iii) Set X in Statement form
X = { prime number less than 13 }

Set Y in Roster form
Y = {2, 3, 5, 7, 11 }

(iv) Set Y in Statement form
Y = { First Five months of a year }

Set Y in roster form
Y = { January , February, March, April, May }

(v) Set Z in Statement form
Z = { Alphabets in word “GOOGLE” }

Set Z in Roster form
Z = { G, O, O, G, L, E }

Eliminating the repeating elements, we get;
Z = { G, O, L, E }

Set Builder form of Set Representation

This form is used when all the elements in the set possess similar property.

Here we define the elements with the help of a formula which is placed inside the curved bracket.

Structure of Set representation:
⟹ We first write a variable using any English Alphabet.
This variable represents the formula of all elements.

⟹ The variable is followed by symbol ” : ” or ” | “
The symbol is read as ” such that”

⟹ Now we write the formula/statement that define all the elements of set

All the above description should be inside the curly bracket of the set.

Examples of Set Builder form


(i) R is a set of number greater than 3 and less than 15

This can be represented in set builder form as:
R = { x : 3 < x < 15 }

The expression is read as;
R is a set of elements x such that the number is greater than 3 and less than 5.

Roster Form
R = { 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

(ii) Set P contains even natural numbers.

Set P represented in set builder form
P = { x : x is a natural number & x = 2n for n \mathtt{\epsilon } N }

The expression is read as follows;
P is a set of element x such that x is a even natural numbers

Note:
The expression x = 2n is a formula for even number.

n \mathtt{\epsilon } N says that n belongs to natural number N.

When you put n = (1, 2, 3, 4 . . . ) in formula x = 2n, you will get even numbers.

x = 2, 4, 6, 8, 10 . . . etc.