Representation of a Set

In representation of a set the following three methods are commonly used:

(i) Statement form method

(ii) Roster or tabular form method

(iii) Rule or set builder form method

1. Statement form:

In this, well-defined description of the elements of the set is given and the same are enclosed in curly brackets.

For example:

(i) The set of odd numbers less than 7 is written as: {odd numbers less than 7}.

(ii) A set of football players with ages between 22 years to 30 years.

(iii) A set of numbers greater than 30 and smaller than 55.

(iv) A set of students in class VII whose weights are more than your weight.

2. Roster form or tabular form:

In this, elements of the set are listed within the pair of brackets { } and are separated by commas.

For example:

(i) Let N denote the set of first five natural numbers.

Therefore, N = {1, 2, 3, 4, 5} → Roster Form

(ii) The set of all vowels of the English alphabet.

Therefore, V = {a, e, i, o, u} → Roster Form

(iii) The set of all odd numbers less than 9.

Therefore, X = {1, 3, 5, 7} → Roster Form

(iv) The set of all natural number which divide 12.

Therefore, Y = {1, 2, 3, 4, 6, 12} → Roster Form

(v) The set of all letters in the word MATHEMATICS.

Therefore, Z = {M, A, T, H, E, I, C, S} → Roster Form

(vi) W is the set of last four months of the year.

Therefore, W = {September, October, November, December} → Roster Form

Note:

The order in which elements are listed is immaterial but elements must not be repeated.

3. Set builder form:

In this, a rule, or the formula or the statement is written within the pair of brackets so that the set is well defined. In the set builder form, all the elements of the set, must possess a single property to become the member of that set.

In this form of representation of a set, the element of the set is described by using a symbol ‘x’ or any other variable followed by a colon The symbol ‘:‘ or ‘|‘ is used to denote such that and then we write the property possessed by the elements of the set and enclose the whole description in braces. In this, the colon stands for ‘such that’ and braces stand for ‘set of all’.

For example:

(i) Let P is a set of counting numbers greater than 12;
the set P in set-builder form is written as :

                P = {x : x is a counting number and greater than 12}
                                                 or
                P = {x | x is a counting number and greater than 12}

This will be read as, 'P is the set of elements x such that x is a counting number and is greater than 12'.

Note:

The symbol ':' or '|' placed between 2 x's stands for such that.

(ii) Let A denote the set of even numbers between 6 and 14. It can be written in the set builder form as;

A = {x|x is an even number, 6 < x < 14}

or A = {x : x ∈ P, 6 < x < 14 and P is an even number}

(iii) If X = {4, 5, 6, 7} . This is expressed in roster form.

Let us express in set builder form.

X = {x : x is a natural number and 3 < x < 8}

(iv) The set A of all odd natural numbers can be written as

A = {x : x is a natural number and x = 2n + 1 for n ∈ W}

Solved example using the three methods of representation of a set:

The set of integers lying between -2 and 3.

Statement form: {I is a set of integers lying between -2 and 3}

Roster form: I = {-1, 0, 1, 2}

Set builder form: I = {x : x ∈ I, -2 < x < 3}