ICSE Class 8 Maths Chapter 37 Sets

Unit - 7: Set Theory

Chapter 37: Sets

37.1 Review

Set: A set is a collection of well-defined objects.

1. The collection of tall students of your class is not well-defined, so it does not form a set.

2. The collection of students of your class with heights between 135 cm and 160 cm is well-defined, so it forms a set.

Elements: The objects (numbers, names, etc.) used to form a set is called elements or members of the set.

In general, a set is represented by capital letters of English alphabet. The elements of the set are written inside curly braces and separated by commas.

(i) If A is the set of names: John, Geeta, Amit and Rohit; then: set A = {John, Geeta, Amit, Rohit}.

(ii) If V is the set of vowels of English alphabet then, set V = {a, e, i, o, u}.

Using 'ε' and ∉: The symbol 'ε' stands for 'belongs to' and the symbol '∉' stands for 'does not belong to'.

If an element x belongs to set A, we write: x ∈ A and if 'x' does not belong to set A, we write x ∉ A.

37.2 Representation Of A Set

There are mainly two ways of representing a set.

(i) Roster or Tabular Form (ii) Rule Method or Set-Builder Form.

1. Roster (or Tabular) Form

In this form, the elements of the set are enclosed in curly braces { } after separating them by commas.

For example, if a set A consists of numbers 2, 5, 7, 9 and 15, it is written as: A = {2, 5, 7, 9, 15}.

More examples:

(i) The set of integers i.e. Z = {.....-, -2, -1, 0, 1, 2, 3, .......}

(ii) The set of whole numbers i.e. W = {0, 1, 2, 3, 4, .........}

(iii) The set of natural numbers i.e. N = {1, 2, 3, 4, .......}

1. The order in which the elements of a set are written is not important. i.e. {a, b, c}, {b, a, c} and {c, b, a} represent the same set.

2. An element of a set is written only once. i.e. (i) {2, 3, 3, 2, 4, 2} = {2, 3, 4} (ii) The set of letters in the word ALLAHABAD = {a, l, h, b, d}

Teacher's Note

Sets help us organize items in real life - like grouping students by height ranges or collecting coins by denomination. Understanding roster form makes it easy to list and count items systematically.

2. Set-Builder Form (Rule Method)

In this form, the actual elements of the set are not written, but a statement or a formula or a rule is written in the briefest possible way to represent the elements of the set.

e.g. Let A be the set of natural numbers less than 7, then in set-builder form it is written as:

A = { x : x ∈ N and x < 7 }

and is read as "A is the set of x such that x is a natural number and x is less than 7."

The symbol ':' is read as such that.

More examples:

1. A = {2, 3, 4, 5} [Roster or Tubular Form]

= {x : x ∈ N, 2 ≤ x < 6} [Set-builder Form]

For x representing the natural numbers 2, 3, 4 and 5; we can also write 1 < x < 6 or 2 ≤ x ≤ 5 or 1 < x ≤ 5.

2. C = {1, 3, 5, 7, 9, 11} [Roster Form]

= {x : x = 2n - 1, n ∈ N and n ≤ 6}

or {x : x = 2n + 1, n ∈ W and n ≤ 5} [Set-builder Form]

3. D = {x : x = 2n, n ∈ N and n ≤ 4} [Set-builder Form]

= {2 × 1, 2 × 2, 2 × 3, 2 × 4}

= {2, 4, 6, 8} [Roster Form]

Test Yourself

1. Set of letters of the word JALLANDHAR = ..........................................

2. Roster form of set A = { 1/2, 2/3, 3/4, 4/5, 5/6, .......} Set-builder form of set A = ............................................

3. Set-builder form of set B = {x : x = n\(^2\), n ∈ W and 4 < n ≤ 8} Roster form of set B = ..................................................

4. 5x - 3 ≥ 12 and x ∈ N ⇒ x = ........., ........., ........., ........., ........., ........., ........., ........., ........., .........,

and A = {x : 5x - 3 ≥ 12 and x ∈ N} = ............................................

Example 1:

Write the following sets in the set-builder form:

(i) {x : x = \(\frac{2n}{n+2}\), n ∈ W and n < 3} (ii) {x : x = 5y - 3, y ∈ Z and -2 ≤ y < 2}

(iii) {x : x ∈ W and 8x + 5 < 23}

Solution:

(i) Since, n ∈ W and n < 3 ⇒ n = 0, 1 and 2

∴ Set in set-builder form = { \(\frac{2 \times 0}{0 + 2}, \frac{2 \times 1}{1 + 2}, \frac{2 \times 2}{2 + 2}\) } = {0, \(\frac{2}{3}\), 1} (Ans.)

(ii) y ∈ Z and -2 ≤ y < 2 ⇒ y = -2, -1, 0 and 1

∴ Set in set-builder form = {5 × -2 - 3, 5 × -1 -3, 5 × 0 - 3, 5 × 1 - 3}

= {-13, -8, -3, 2} (Ans.)

(iii) 8x + 5 < 23 ⇒ 8x < 18 and x < 2.25

∴ x ∈ W and x < 2.25 ⇒ x = 0, 1 and 2

∴ Set in set-builder form = {0, 1, 2} (Ans.)

Teacher's Note

Set-builder notation is like creating a rule to describe a group - similar to how a dress code describes what students should wear rather than listing each student's outfit.

Example 2:

Express the following sets in set-builder form:

(i) {\(\frac{7}{8}, \frac{8}{9}, \frac{9}{10}, \frac{10}{11}, \frac{11}{12}\)} (ii) {0, 3, 6, 9, 12, 15, 18}

(iii) {\(\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}\)} (iv) {x : x\(^2\) - 6x - 7 = 0}

Solution:

(i) {\(\frac{7}{8}, \frac{8}{9}, \frac{9}{10}, \frac{10}{11}, \frac{11}{12}\)} = {\(\frac{7}{7+1}, \frac{8}{8+1}, \frac{9}{9+1}, \frac{10}{10+1}, \frac{11}{11+1}\)}

= {x : x = \(\frac{n}{n+1}\), n ∈ N and 7 ≤ n ≤ 11} (Ans.)

(ii) {0, 3, 6, 9, 12, 15, 18} = {3 × 0, 3 × 1, 3 × 2, 3 × 3, 3 × 4, 3 × 5, 3 × 6}

= {x : x = 3n, n ∈ W and x ≤ 6} (Ans.)

(iii) {\(\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}\)} = {\(\frac{1}{3^1}, \frac{1}{3^2}, \frac{1}{3^3}, \frac{1}{3^4}, \frac{1}{3^5}\)}

= {x : x = \(\frac{1}{3^n}\), n ∈ N and n ≤ 5} (Ans.)

(iv) Since, x\(^2\) - 6x - 7 = 0

⇒ x\(^2\) - 7x + x - 7 = 0 i.e. x(x - 7) + 1(x - 7) = 0

⇒ (x - 7) (x + 1) = 0 i.e. x - 7 = 0 or x + 1 = 0

⇒ x = 7 or x = -1

∴ {x : x\(^2\) - 6x - 7 = 0} = {7, -1} (Ans.)

Teacher's Note

Understanding how to write sets in different forms is like learning two languages for the same concept - one shows all the items (roster) and one gives you the rule to find them (set-builder).

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