Chapter 3

Exercise 3.1 Solutions

Clean, step-by-step solved exercises formatted with premium math typesetting using native HTML/CSS layout.

Question 1 Set Builder Notation
Write the following sets in builder notation:

(i) {1, 4, 9, 16, 25, 36, ..., 484}

Solution
Observe that each element is a perfect square: 12 = 1, 22 = 4, 32 = 9, ..., 222 = 484.
The numbers being squared are natural numbers from 1 to 22.
Set builder notation: {x | x = n2, n ∈ N and 1 ≤ n ≤ 22}

(ii) {2, 4, 8, 16, 32, ..., 1024}

Solution
Observe that each element is a power of 2: 21 = 2, 22 = 4, 23 = 8, ..., 210 = 1024.
The powers are natural numbers from 1 to 10.
Set builder notation: {x | x = 2n, n ∈ N and 1 ≤ n ≤ 10}

(iii) {0, ±1, ±2, ..., ±1000}

Solution
These are all integers from -1000 to 1000.
Set builder notation: {x | x ∈ Z and -1000 ≤ x ≤ 1000}

(iv) {6, 12, 18, ..., 120}

Solution
Observe that each element is a multiple of 6: 6 × 1 = 6, 6 × 2 = 12, ..., 6 × 20 = 120.
Set builder notation: {x | x = 6n, n ∈ N and 1 ≤ n ≤ 20}

(v) {100, 102, 104, ..., 400}

Solution
These are consecutive even natural numbers from 100 to 400.
Set builder notation: {x | x = 2n, n ∈ N and 50 ≤ n ≤ 200}

(vi) {1, 3, 9, 27, 81, ...}

Solution
Observe that each element is a power of 3 starting from 0: 30 = 1, 31 = 3, 32 = 9, ...
Set builder notation (using Whole Numbers W): {x | x = 3n, n ∈ W}

(vii) {1, 2, 4, 5, 10, 20, 25, 50, 100}

Solution
Observe that these numbers are factors (divisors) of 100.
Set builder notation: {x | x is a divisor of 100}

(viii) {5, 10, 15, ..., 100}

Solution
These are multiples of 5 from 5 × 1 up to 5 × 20.
Set builder notation: {x | x = 5n, n ∈ N and 1 ≤ n ≤ 20}

(ix) The set of all integers between -100 and 1000

Solution
Set builder notation: {x | x ∈ Z and -100 < x < 1000}
Question 2 Tabular Form
Write each of the following sets in tabular forms:

(i) {x | x is a multiple of 3 and x ≤ 36}

Solution
The multiples of 3 less than or equal to 36 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36.
Tabular form: {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36}

(ii) {x ∈ R | 2x + 1 = 0}

Solution
Solve the equation:
2x + 1 = 0 2x = -1 x = -12
Since -1/2 is a real number (-1/2 ∈ R):
Tabular form: {-1/2}

(iii) {x ∈ P | x < 12} (where P represents prime numbers)

Solution
Prime numbers (P) are numbers with exactly two divisors (1 and itself). Prime numbers less than 12 are: 2, 3, 5, 7, 11.
Tabular form: {2, 3, 5, 7, 11}

(iv) {x | x is a divisor of 128}

Solution
The positive numbers that divide 128 without remainder are: 1, 2, 4, 8, 16, 32, 64, 128.
Tabular form: {1, 2, 4, 8, 16, 32, 64, 128}

(v) {x | x = 2n, n ∈ N and n < 8}

Solution
Since n ∈ N and n < 8, the possible values of n are 1, 2, 3, 4, 5, 6, 7.
Calculate 2n for each:
n = 1 ⇒ 21 = 2 n = 2 ⇒ 22 = 4 n = 3 ⇒ 23 = 8 n = 4 ⇒ 24 = 16 n = 5 ⇒ 25 = 32 n = 6 ⇒ 26 = 64 n = 7 ⇒ 27 = 128
Tabular form: {2, 4, 8, 16, 32, 64, 128}

(vi) {x ∈ N | x + 4 = 0}

Solution
Solve the equation: x + 4 = 0 ⇒ x = -4.
Since -4 is not a natural number (-4 ∉ N), there is no element satisfying this property.
Tabular form: { } (Empty set or ∅)

(vii) {x ∈ N | x = x}

Solution
The property x = x is true for all natural numbers.
Tabular form: {1, 2, 3, 4, 5, ...}

(viii) {x ∈ Z | 3x + 1 = 0}

Solution
Solve the equation: 3x + 1 = 0 ⇒ 3x = -1 ⇒ x = -1/3.
Since -1/3 is not an integer (-1/3 ∉ Z), no element satisfies the condition.
Tabular form: { } (Empty set or ∅)
Question 3 Proper Subsets
Write two proper subsets of each of the following sets:

(i) {a, b, c}

Solution
Proper subsets of {a, b, c} must contain elements from the set but cannot be equal to the set itself.
Two proper subsets: {a} and {b} (or {a, b})

(ii) {0, 1}

Solution
Two proper subsets: {0} and {1} (or the empty set ∅)

(iii) N (Set of Natural Numbers)

Solution
Natural numbers: N = {1, 2, 3, 4, ...}.
Two proper subsets: {1, 3, 5, 7, ...} (Odd natural numbers) and {2, 4, 6, 8, ...} (Even natural numbers)

(iv) Z (Set of Integers)

Solution
Integers: Z = {0, ±1, ±2, ±3, ...}.
Two proper subsets: N (Set of Natural numbers) and W (Set of Whole numbers)

(v) Q (Set of Rational Numbers)

Solution
Two proper subsets: N (Natural numbers) and Z (Integers)

(vi) R (Set of Real Numbers)

Solution
Two proper subsets: Q (Rational numbers) and Q' (Irrational numbers)

(vii) {x ∈ Q | 0 < x < 2}

Solution
Any set containing values strictly between 0 and 2 that are rational is a proper subset.
Two proper subsets: {1} and {0.5, 1.5}
Question 4 Empty Set Property
Is there any set which has no proper subset? If so, name that set.
Solution
Yes, there is such a set. It is the Empty Set (null set), denoted by { } or ∅.
Explanation: The only subset of the empty set is itself ∅. A proper subset of a set A must be a subset of A that is not equal to A. Since there are no other subsets, the empty set has no proper subset.
Question 5 Set Differences
What is the difference between {a, b} and {{a, b}}?
Solution
{a, b} is a set containing exactly two distinct elements, namely a and b. Its cardinality is 2.
{{a, b}} is a singleton set (a set containing only one element). Its single element is itself the set {a, b}. Its cardinality is 1.
Question 6 Power Set Cardinality
What is the number of elements of the power set of each of the following sets?

(i) { }

Solution
Number of elements in the set: n = 0.
Number of elements in its power set: 2n = 20 = 1.

(ii) {0, 1}

Solution
Number of elements in the set: n = 2.
Number of elements in its power set: 2n = 22 = 4.

(iii) {1, 2, 3, 4, 5, 6, 7}

Solution
Number of elements in the set: n = 7.
Number of elements in its power set: 2n = 27 = 128.

(iv) {0, 1, 2, 3, 4, 5, 6, 7}

Solution
Number of elements in the set: n = 8.
Number of elements in its power set: 2n = 28 = 256.

(v) {a, {b, c}}

Solution
The elements of this set are: a and {b, c}.
Thus, the number of elements in the set is: n = 2.
Number of elements in its power set: 2n = 22 = 4.

(vi) {{a, b}, {b, c}, {d, e}}

Solution
The elements of this set are three subsets: {a, b}, {b, c}, and {d, e}.
Thus, the number of elements in the set is: n = 3.
Number of elements in its power set: 2n = 23 = 8.
Question 7 Power Set Elements
Write down the power set of each of the following sets:

(i) {9, 11}

Solution
Given set: S = {9, 11}, elements count: n = 2.
The power set P(S) must have 22 = 4 elements.
Power set: P(S) = {∅, {9}, {11}, {9, 11}}

(ii) {+, -, ×, ÷}

Solution
Given set: S = {+, -, ×, ÷}, elements count: n = 4.
The power set P(S) must have 24 = 16 elements.
Power set: P(S) = {∅, {+}, {-}, {×}, {÷}, {+, -}, {+, ×}, {+, ÷}, {-, ×}, {-, ÷}, {×, ÷}, {+, -, ×}, {+, -, ÷}, {+, ×, ÷}, {-, ×, ÷}, {+, -, ×, ÷}}

(iii) {∅}

Solution
Given set: S = {∅}. It contains 1 element (which is the empty set itself). Thus n = 1.
The power set P(S) must have 21 = 2 elements.
Power set: P(S) = {∅, {∅}}

(iv) {a, {b, c}}

Solution
Given set: S = {a, {b, c}}, elements count: n = 2.
The power set P(S) must have 22 = 4 elements.
Power set: P(S) = {∅, {a}, {{b, c}}, {a, {b, c}}}
Interactive Sandbox Explore Set Builder & Power Sets
Configure set-builder properties below to visually generate sets in tabular form, or input a custom set to list its subsets and build its power set in real-time.
Set Builder Notation
Tabular Form
Power Set Generator
Cardinality & Set Details
Proper Subsets
Power Set P(S)