Exercise 2.1

1. Which of the following are sets? Justify your answer.

(i) The collection of all the months of a year beginning with the letter "J".

Solution:

This is a .

Justification: The collection is well-defined because we can clearly identify which months begin with "J".

Months beginning with "J": , , (enter in order of occurence)

Since the elements are and , this forms a valid set.

(ii) The collection of ten most talented writers of India.

Solution:

This .

Justification: The collection is not because:

The term "most talented" is

Different people may have different opinions about who are the most talented writers. There is no objective criteria to determine the "ten most talented".

For a collection to be a set, the membership criteria must be clear and unambiguous.

(iii) A team of eleven best cricket batsmen of the world.

Solution:

This is .

Justification: The collection because:

• The term "best" is

• Different cricket experts may choose different players. There is no universal standard to determine who are the "best" batsmen. So, the criteria for membership is not objective and precise.

(iv) The collection of all boys in your class.

Solution:

This is .

Justification: The collection is because:

• We can clearly identify who are the in the class.

• We can clearly identify which is being referred to.

• Each student is either a boy or not a boy - there is no ambiguity. The membership criteria is clear and objective.

(v) The collection of all even integers.

Solution:

This is .

Justification: The collection is because:

• An even integer is any integer divisible by .

• For any integer, we can determine whether it is or not.

• The definition is mathematical and precise. Examples: ..., -4, -2, 0, 2, 4, 6, 8, ...

2. If A = {0, 2, 4, 6}, B = {3, 5, 7} and C = {p, q, r}, then fill the appropriate symbol, ∈ or ∉ in the blanks.

(i) 0 A since 0 is present in set A.

(ii) 3 C since 3 is not present in set C.

(iii) 4 B since 4 is not present in set B.

(iv) 8 A since 8 is not present in set A.

(v) p C since p is present in set C.

(vi) 7 B since 7 is present in set B.

3. Express the following statements using symbols.

(i) The element 'x' does not belong to 'A'.

Solution: Using set notation symbols: x A

The symbol ∉ means "does not belong to" or "is not an element of".

(ii) 'd' is an element of the set 'B'.

Solution: Using set notation symbols: d B

The symbol ∈ means "is an element of" or "belongs to".

(iii) '1' belongs to the set of Natural numbers.

Solution: Let N represent the set of natural numbers. Using set notation symbols: 1 N

Since 1 a natural number, it to the set N.

(iv) '8' does not belong to the set of prime numbers P.

Solution: Using set notation symbols: 8 P

8 a prime number because it has factors other than 1 and itself (factors: , , , ).

4. State whether the following statements are true or false. Justify your answer.

5 ∉ set of prime numbers

S = {5, 6, 7} implies 8 ∈ S

-5 ∉ W where 'W' is the set of whole numbers

8/11 ∉ Z, where 'Z' is the set of integers

True

False

Justification

(i) 5 ∉ set of prime numbers: 5 is a number as it has exactly factors: and

(ii) S = {5, 6, 7} implies 8 ∈ S: Set S = {5, 6, 7} contains only elements: , , and . Since is not listed as an element of set S, we have 8 ∉ S.

(iii) -5 ∉ W where 'W' is the set of whole numbers: Whole numbers include all integers starting from .

(iv) 811 ∉ Z, where 'Z' is the set of integers: Integers are numbers and their negatives while 811 is a (rational number) that cannot be simplified to an integer.

5. Write the following sets in roster form.

(i) B = {x : x is a natural number smaller than 6}

Solution: Natural numbers are: 1, 2, 3, 4, 5, 6, 7, ... Natural numbers smaller than 6 are: , , , ,

Therefore, B = {1, 2, 3, 4, 5}

(ii) C = {x : x is a two digit natural number such that the sum of its digits is 8}

Solution: Two-digit numbers range from to . We need numbers where the sum of digits equals : (1) 17: 1 + 7 = (2) 26: 2 + 6 = (3) 35: 3 + 5 =

(4) 44: 4 + 4 = (5) 53: 5 + 3 = (6) 62: 6 + 2 = (7) 71: 7 + 1 = (8) 80: 8 + 0 =

Therefore, C = {17, 26, 35, 44, 53, 62, 71, 80}

(iii) D = {x : x is a prime number which is a divisor of 60}

Solution: First, find all divisors of 60: 60 = 22 × 3 × 5 Divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 Among these, the prime numbers are: , ,

Therefore, D = {2, 3, 5}

(iv) E = {x : x is an alphabet in BETTER}

Solution: The word "BETTER" contains the letters: B, E, T, T, E, R Distinct letters (removing repetitions): , , ,

Therefore, E = {B, E, T, R}

6. Write the following sets in the set-builder form.

(i) {3, 6, 9, 12}

Solution: Observing the pattern: 3, 6, 9, 12. These are multiples of : 3×1, 3×2, 3×3, 3×4 Set-builder form: {x : x = , n ∈ {1, 2, 3, 4}}

Alternative form: {x : x is a multiple of 3 and }

(ii) {2, 4, 8, 16, 32}

Solution: Observing the pattern: 2, 4, 8, 16, 32 These are powers of : 21, 22, 23, 24, 25 Set-builder form: { x : x = , n ∈ {1, 2, 3, 4, 5} }

Alternative form: { x : x is a power of 2 and x }

(iii) {5, 25, 125, 625}

Solution: Observing the pattern: 5, 25, 125, 625 These are powers of : 51, 52, 53, 54 Set-builder form: {x : x = , n ∈ {1, 2, 3, 4}}

Alternative form: {x : x is a power of 5 and x }

(iv) {1, 4, 9, 16, 25, ..., 100}

Solution: Observing the pattern: 1, 4, 9, 16, 25, ..., 100 These are perfect squares: 12, 22, 32, 42, 52, ..., 102 Set-builder form: {x : x = , n ∈ N and

}

Alternative form: {x : x is a perfect square and x }

7. Write the following sets in roster form.

Write set A in roster form: A = natural numbers greater than 50 but smaller than 100.

Natural numbers are positive integers: 1, 2, 3, 4, ... We need numbers where 50 < n < 100. The first number greater than 50 is , and the last number smaller than 100 is .

Solution: A = {51, 52, 53, 54, ..., 98, 99}

Write set B in roster form: B = integers where x2 = 4.

We need to solve x2 = 4. Taking the square root of both sides: x = ±4 = ±. So we have two solutions: x = 2 and x = -2.

Solution: B = {-2, 2}

Write set D in roster form: D = letters in word "LOYAL".

List each distinct letter that appears in "LOYAL". The letters are: L, O, Y, A, L. Since L appears twice, we only list it in the set.

Solution: D = {L, O, Y, A}

Write set E in roster form: E = x = 2n2 + 1 for -3 ≤ n ≤ 3.

Calculate x = 2n2 + 1 for each integer n from -3 to 3: (i) n = -3: x = 2−32 + 1 = 2(9) + 1 = ; (ii) n = -2: x = 2−22 + 1 = 2(4) + 1 = ; (iii) n = -1: x = 2−12 + 1 = 2(1) + 1 =

(iv) n = 0: x = 202 + 1 = 2(0) + 1 = ; (v) n = 1: x = 212 + 1 = 2(1) + 1 = ; (vi) n = 2: x = 222 + 1 = 2(4) + 1 = ; (vii) n = 3: x = 232 + 1 = 2(9) + 1 =

Since sets contain only distinct elements, we list each value once.

Solution: E = {1, 3, 9, 19}

Chapter 2: Sets

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Exercise 2.1

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Chapter 2: Sets

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Glossary

Exercise 2.1