Sets - Exercise 6(B)

Indicate whether the given statement is true or false :

(i) {Triangles} ⊆ {Quadrilaterals}

(ii) {Squares} ⊆ {Rectangles}

(iii) {Rhombuses} ⊆ {Parallelograms}

(iv) {Natural numbers} ⊆ {Whole numbers}

(v) {Integers} ⊆ {Whole numbers}

(vi) {Composite numbers} ⊆ {Odd numbers}

Answer

(i) False
Reason — A triangle is a polygon with 3 sides, whereas a quadrilateral is a polygon with 4 sides. Since no triangle can be a quadrilateral, the set of triangles is not a subset of the set of quadrilaterals.

(ii) True
Reason — A square is defined as a special type of rectangle where all four sides are equal. Since every square satisfies the properties of a rectangle, the set of squares is a subset of the set of rectangles.

(iii) True
Reason — A rhombus is a quadrilateral with both pairs of opposite sides parallel and all sides equal. Since it satisfies the definition of a parallelogram (a quadrilateral with two pairs of parallel sides), the set of rhombuses is a subset of the set of parallelograms.

(iv) True
Reason — Natural numbers (N) are {1, 2, 3, ...} and whole numbers (W) are {0, 1, 2, 3, ....}. Since every natural number is also a whole number, {Natural numbers} ⊆ {Whole numbers}.

(v) False
Reason — Integers include both, negative and positive numbers (...., -2, -1, 0, 1, 2, ....), whereas whole numbers consist only of zero and positive counting numbers. Since negative integers are not whole numbers, the set of integers is not a subset of whole numbers.

(vi) False
Reason — Composite numbers are numbers with more than two factors, such as 4, 6, 8, 9, 10, ... . Many composite numbers (like 4, 6, and 8) are even, so the set of composite numbers is not a subset of the set of odd numbers.

Write down all possible subsets of each of the sets given below :

(i) {1}

(ii) {3, 4}

(iii) {2, 3, 5}

(iv) Φ

(v) {c, d, e}

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

Answer

(i) {1}

Subsets are: Φ, {1}

(ii) {3, 4}

Subsets are: Φ, {3}, {4}, {3, 4}

(iii) {2, 3, 5}

Subsets are: Φ, {2}, {3}, {5}, {2, 3}, {3, 5}, {2, 5}, {2, 3, 5}

(iv) Φ

The empty set has only itself as a subset.

Subsets are: Φ

(v) {c, d, e}

Subsets are: Φ, {c}, {d}, {e}, {c, d}, {d, e}, {c, e}, {c, d, e}

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

Subsets are: Φ, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}

Write down all possible proper subsets of each of the sets given below :

(i) {x}

(ii) {p, q}

(iii) {m, n, p}

(iv) {1, 2, 3, 4}

Answer

(i) {x}

Proper Subset is Φ

(ii) {p, q}

Proper Subsets are: Φ, {p}, {q}

(iii) {m, n, p}

Proper Subsets are: Φ, {m}, {n}, {p}, {m, n}, {n, p}, {m, p}

(iv) {1, 2, 3, 4}

Proper Subsets are: Φ, {1}, {2}, {3}, {4} {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}

Write down :

(i) The set C of letters of the word 'PAPAYA'.

(ii) All subsets of C.

(iii) All proper subsets of C.

Answer

(i) The set C of letters of the word 'PAPAYA'.

In roster form, every repeated element in a set is taken only once.

The distinct letters in the word 'PAPAYA' are P, A, and Y.

C = {P, A, Y}

(ii) All subsets of C.

The empty set (Φ) is a subset of every set.

Every set is a subset of itself.

Since set C has 3 elements, it will have 2³ = 8 subsets.

Subsets are: Φ, {P}, {A}, {Y}, {P, A}, {A, Y}, {P, Y}, {P, A, Y}

(iii) All proper subsets of C.

A proper subset includes all subsets of the set except for the set itself.

There are 2ⁿ - 1 proper subsets, which means 8 - 1 = 7 for this set.

Proper subsets are: Φ, {P}, {A}, {Y}, {P, A}, {A, Y}, {P, Y}

How many subsets in all are there of a set containing 4 elements?

Answer

Number of elements (n) = 4.

Formula for total number of subsets = 2ⁿ.

By replacing 'n' with 4, we get:

2⁴ = 2 x 2 x 2 x 2 = 16.

Hence, there are 16 subsets in all.

How many subsets in all are there of a set with cardinal number 6?

Answer

Cardinal number (n) = 6.

Formula for total number of subsets = 2ⁿ.

By replacing 'n' with 6, we get:

2⁶ = 2 x 2 x 2 x 2 x 2 x 2 = 64.

Hence, there are 64 subsets in all.

How many proper subsets in all are there of a set containing 3 elements?

Answer

Number of elements (n) = 3.

Formula for number of proper subsets = 2ⁿ - 1

By replacing 'n' with 3, we get:

2³ - 1 ⇒ 8 - 1 ⇒ 7

Hence, there are 7 proper subsets in all.

How many proper subsets in all are there of a set with cardinal number 5?

Answer

Cardinal number (n) = 5.

Formula for number of proper subsets = 2ⁿ - 1

By replacing 'n' with 5, we get:

2⁵ - 1 ⇒ 32 - 1 ⇒ 31

Hence, there are 31 proper subsets in all.

Which of the following statements are true ?

(i) {a} ⊂ {a, b, c}

(ii) {a} ⊂ {b, c, d, e}

(iii) Φ ⊂ {a, b, c}

(iv) Φ ∈ {a, b, c}

(v) 0 ∉ Φ

(vi) {1} ⊂ {0, 1}

(vii) Every subset of a finite set is finite.

(viii) Every subset of an infinite set is infinite.

Answer

(i) True
Reason — The element 'a' is present in the set {a, b, c}. Since {a} is a set containing an element from the second set, it is a proper subset.

(ii) False
Reason — For {a} to be a subset of {b, c, d, e}, the element 'a' must be present in the second set. Since it is not, the statement is false.

(iii) True
Reason — By definition, the empty set (Φ) is a subset of every set.

(iv) False
Reason — The symbol ∈ means "is an element of." Φ is a subset of {a, b, c}, not an element of it.

(v) True
Reason — The empty set (Φ) contains no elements at all. Therefore, it is correct to say that 0 is not an element of Φ.

(vi) True
Reason — The element 1 is present in the set {0, 1}, making {1} a proper subset.

(vii) True
Reason — A finite set has a specific number of elements. Any collection of elements taken from it will also have a specific, countable number of elements.

(viii) False
Reason — While an infinite set has endless elements, you can still pick a limited number of elements from it to form a subset. For example, {1, 2} is a finite subset of the infinite set of natural numbers {1, 2, 3, ....}.

Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

Write the subset of A containing :

(i) all odd numbers

(ii) all prime numbers

(iii) all multiples of 4.

Answer

(i) all odd numbers

Given A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

Odd numbers are: {1, 3, 5, 7, 9, 11}

(ii) all prime numbers

Prime numbers are natural numbers greater than 1 that have exactly two factors: 1 and the number itself. From set A, these are 2, 3, 5, 7, and 11.

All prime numbers are: {2, 3, 5, 7, 11}

(iii) all multiples of 4.

Multiples of 4 are numbers that can be divided by 4 without a remainder. Within the range of set A, these are 4 x 1, 4 x 2, and 4 x 3.

All multiples of 4 are: {4, 8, 12}

Let U = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} be the universal set and let A = {5, 7, 11, 13}, B = {6, 8, 10, 12, 14, 16} and C = {5, 6, 8, 10, 11, 12} be its subsets.

Find:

(i) A'

(ii) B'

(iii) C'

Answer

(i) A'

Universal Set (U) = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

Set A = {5, 7, 11, 13}

Remove 5, 7, 11, and 13 from U.

A' = {6, 8, 9, 10, 12, 14, 15, 16}

(ii) B'

Universal Set (U) = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

B = {6, 8, 10, 12, 14, 16}

Remove 6, 8, 10, 12, 14, and 16 from U.

B' = {5, 7, 9, 11, 13, 15}

(iii) C'

Universal Set (U) = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

C = {5, 6, 8, 10, 11, 12}

Remove 5, 6, 8, 10, 11, and 12 from U.

C' = {7, 9, 13, 14, 15, 16}

Let the set I of all integers be the universal set and let A = {x : x is a negative integer} be its subset. Find A'.

Answer

Universal Set (I) = The set of all integers, which includes negative integers, zero, and positive integers. I = {..., -2, -1, 0, 1, 2, ....}.

A = The set of all negative integers = {..., -3, -2, -1}.

Complement (A') = This set consists of all integers that are not negative. This includes zero and all positive integers (0, 1, 2, 3, ....).

∴ A' = {x : x is a non-negative integer}

Suggest a universal set for the sets given below :

(i) {5, 7, 9}, {3, 5, 7}, {1, 3, 9} and {2, 4, 8}.

(ii) {odd numbers less than 8}, {prime numbers less than 8} and {even numbers between 3 and 8}.

(iii) {vowels in English alphabet}, {consonants in English alphabet}.

Answer

(i) {5, 7, 9}, {3, 5, 7}, {1, 3, 9} and {2, 4, 8}.

Subsets = {5, 7, 9}, {3, 5, 7}, {1, 3, 9} and {2, 4, 8}.

The elements present across all sets are 1, 2, 3, 4, 5, 7, 8, and 9.

∴ Universal Set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

(ii) {odd numbers less than 8}, {prime numbers less than 8} and {even numbers between 3 and 8}.

Subsets = {odd numbers less than 8}, {prime numbers less than 8} and {even numbers between 3 and 8}

Odd numbers less than 8 = {1, 3, 5, 7}

Prime numbers less than 8 = {2, 3, 5, 7}

Even numbers between 3 and 8 = {4, 6}

The elements present across all sets are 1, 2, 3, 4, 5, 6, 7.

∴ Universal Set U = {1, 2, 3, 4, 5, 6, 7}

(iii) {vowels in English alphabet}, {consonants in English alphabet}.

Subsets = {vowels in English alphabet}, {consonants in English alphabet}.

The first set contains {a, e, i, o, u} and the second contains all other letters of the alphabet.

∴ Universal Set U = {x : x is a letter in English alphabet}.