Class 8 Readmore Maths Solution | Chapter 1 Sets

Chapter 1 Sets

(1) A set is a collection of distinct objects.

(2) A set is said to be finite if it has finite number of elements.

(3) A set is said to be infinite if it has infinite number of elements.

(4) A set that contains no elements is called a null or empty set.

(5) A set that contains only one element is called a singleton set.

(6) Two sets are equal if they have exactly the same elements, regardless of their order of listing.

(7) A set that contains all possible elements is called a universal set.

(8) One of the attractive and useful methods of representing sets is by the means of diagram which is called Venn diagram.

(9) Two or more than two sets are said to be overlapping sets if they have at least one common element and at least one distinct element.

(10) The sets A and B are called disjoint sets if there is no any element in common.

EXERCISE 1. 1

Question 1: Define the following: (a) Finite Set (b) Infinite Set (c) Null Set / Empty Set (d) Singleton Set (e) Equal Sets (f) Disjoint Sets

Solution:

(a) Finite Set

A finite set is a set that has a limited or countable number of elements.

(b) Infinite Set

An infinite set is a set that has an unlimited or uncountable number of elements.

(c) Null Set / Empty Set

A null set or empty set is a set that contains no elements, denoted as {} or ∅.

(d) Singleton Set

A singleton set is a set that contains exactly one element.

(e) Equal Sets

Two sets are equal if they contain exactly the same elements, regardless of order.

(f) Disjoint Sets

Two sets are disjoint if they have no elements in common, i.e., their intersection is the empty set.

Question 2: What do you understand by the overlapping sets?

Overlapping sets are sets that have at least one element in common, i.e., their intersection is not empty.

Question 3: What type of sets are A = {2, 4, 6, 8} and B = {2, 5, 6}?

Solution:

A = {2, 4, 6, 8}

B = {2, 5, 6}

A ∩ B = {2, 6} (common elements).

Since A ∩ B is not empty, A and B are overlapping sets.

Therefore, A and B are overlapping sets.

Question 4: If X = {a, b, c, d} and Y = {x, y, z}, then what type of sets are X and Y?

Solution:

X = {a, b, c, d}

Y = {x, y, z}

X ∩ Y = {} (no common elements).

Since X ∩ Y is empty, X and Y are disjoint sets.

Therefore, X and Y are disjoint sets.

Question 5: If A = {1, 2, 3, 4}, B = {4, 5, 6}, and C = {5, 6, 7, 8}, which set is the overlapping set of set A?

Solution:

A = {1, 2, 3, 4}

B = {4, 5, 6}

C = {5, 6, 7, 8}

A set is overlapping with A if it has at least one element in common with A.

A ∩ B = {4} (not empty).

A ∩ C = {} (empty).

Therefore, B is the overlapping set of A.

Question 6: If A = {a, b, c, d}, B = {e, i, u}, and C = {c, d, e}, which set is the overlapping set of set A?

Solution:

A = {a, b, c, d}

B = {e, i, u}

C = {c, d, e}

A set is overlapping with A if it has at least one element in common with A.

A ∩ B = {} (empty).

A ∩ C = {c, d} (not empty).

Therefore, C is the overlapping set of A.

Question 7: If P = {1, 2, 3}, Q = {4, 5, 6}, and R = {1, 3, 5}, which set is the disjoint set of P?

Solution:

P = {1, 2, 3}

Q = {4, 5, 6}

R = {1, 3, 5}

A set is disjoint with P if it has no elements in common with P.

P ∩ Q = {} (empty).

P ∩ R = {1, 3} (not empty).

Therefore, Q is the disjoint set of P.

Question 8: If K = {i, j, k}, P = {m, n, o}, and R = {n, w, x}, which set is the disjoint set of R?

Solution:

K = {i, j, k}

P = {m, n, o}

R = {n, w, x}

A set is disjoint with R if it has no elements in common with R.

R ∩ K = {} (empty).

R ∩ P = {n} (not empty).

Therefore, K is the disjoint set of R.

SOLVE

Question 1: State overlapping or disjoint sets in the pair of sets from the following:

(a) A = {even numbers less than 10}, B = {2, 4, 5, 6}

Solution:

A = {2, 4, 6, 8}

B = {2, 4, 5, 6}

Common elements: {2, 4, 6}

Since there are common elements, A and B are overlapping sets.

(b) C = {natural numbers less than 5}, D = {cube numbers less than 20}

Solution:

C = {1, 2, 3, 4}

D = {1, 8} (since 1³ = 1, 2³ = 8, 3³ = 27 which is greater than 20)

Common elements: {1}

Since there is a common element, C and D are overlapping sets.

(c) X = {a, b, c, d}, Y = {x, y, z}

Solution:

X = {a, b, c, d}

Y = {x, y, z}

Common elements: { }

Since there are no common elements, X and Y are disjoint sets.

(d) R = {countries of SAARC}, A = {Nepal, India, Bhutan, China}

Solution:

R = {Nepal, India, Bhutan, Sri Lanka, Maldives, Pakistan, Bangladesh, Afghanistan}

A = {Nepal, India, Bhutan, China}

Common elements: {Nepal, India, Bhutan}

Since there are common elements, R and A are overlapping sets.

(e) M = {multiples of 12}, F₁₂ = {factors of 12}

Solution:

M = {12, 24, 36,...}

F₁₂ = {1, 2, 3, 4, 6, 12}

Common elements: {12}

Since there is a common element, M and F₁₂ are overlapping sets.

(f) D = {Chitwan, Makawanpur, Bara, Parsa, Rautahat}, N = {Ram, Narayan, Yadav}

Solution:

D = {Chitwan, Makawanpur, Bara, Parsa, Rautahat}

N = {Ram, Narayan, Yadav}

Common elements: {}

Since there are no common elements, D and N are disjoint sets.

Question 2: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, list the elements of the following sets and represent in Venn diagram: (a) O = {odd numbers} (b) P = {prime numbers} (c) E = {even numbers} (d) F₆ = {factors of 6} (e) M₂ = {multiples of 2} (f) M₆ = {multiples of 6}

Universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(a) O = {odd numbers}

Solution:

O = {1, 3, 5, 7, 9}

(b) P = {prime numbers}

Solution:

P = {2, 3, 5, 7}

(c) E = {even numbers}

Solution:

E = {2, 4, 6, 8, 10}

(d) F₆ = {factors of 6}

Solution:

F₆ = {1, 2, 3, 6}

(e) M₂ = {multiples of 2}

Solution:

M₂ = {2, 4, 6, 8, 10}

(f) M₆ = {multiples of 6}

Solution:

M₆ = {6}

Question 3: Show the following sets in Venn diagram. Take U as a universal set: (a) A = {2, 4, 6} (b) B = {4, 8, 10} (c) G = {1, 3, 5} (d) H = {a, b, c, d} (e) I = {i, n, k} (f) K = {v, a, m, e, s, h}

(a) A = {2, 4, 6}

Solution:

A = {2, 4, 6}

(b) B = {4, 8, 10}

Solution:

B = {4, 8, 10}

(c) G = {1, 3, 5}

Solution:

G = {1, 3, 5}

(d) H = {a, b, c, d}

Solution:

H = {a, b, c, d}

(e) I = {i, n, k}

Solution:

I = {i, n, k}

(f) K = {v, a, m, e, s, h}

Solution:

K = {v, a, m, e, s, h}

Universal set U is the union of all elements in the given sets:

U={1, 2, 3, 4, 5, 6, 8, 10, a, b, c, d, e, h, i, k, m, n, s, v}

Question 4: If U = {a, b, c, d, e, f, g, x, y, z}, show the following in the Venn diagram.

Solution:

Universal set U = {a, b, c, d, e, f, g, x, y, z}

(a) X = {a, b, c, d}, Y = {b, c, x, y, z}

Solution:

X = {a, b, c, d}

Y = {b, c, x, y, z}

For the Venn diagram:

- X ∩ Y = {b, c} (common elements).

- X - Y = {a, d}, Y - X = {x, y, z}.

- Elements not in X or Y: {e, f, g}.

The Venn diagram shows two overlapping circles: X with {a, b, c, d} and Y with {b, c, x, y, z}, overlapping at {b, c}, and {e, f, g} outside both circles in U.

(b) A = {e, f, g} and B = {x, y, z}

Solution:

A = {e, f, g}

B = {x, y, z}

For the Venn diagram:

- A ∩ B = {} (no common elements).

- Elements not in A or B: {a, b, c, d}.

The Venn diagram shows two separate circles: A with {e, f, g} and B with {x, y, z}, with no overlap, and {a, b, c, d} outside both circles in U.

(c) R = {a, c, f, x, y, z} and Z = {x, y, z}

Solution:

R = {a, c, f, x, y, z}

Z = {x, y, z}

For the Venn diagram:

- R ∩ Z = {x, y, z} (common elements).

- R - Z = {a, c, f}, Z - R = {}.

- Elements not in R or Z: {b, d, e, g}.

The Venn diagram shows two overlapping circles: R with {a, c, f, x, y, z} and Z with {x, y, z}, overlapping at {x, y, z}, and {b, d, e, g} outside both circles in U.

(d) T = {y, z, b} and S = {a, b, y, z}

Solution:

T = {y, z, b}

S = {a, b, y, z}

For the Venn diagram:

- T ∩ S = {y, z, b} (common elements).

- T ⊆ S (T is a subset of S).

- S - T = {a}, T - S = {}.

- Elements not in T or S: {c, d, e, f, g, x}.

The Venn diagram shows two circles: S with {a, b, y, z} and T inside S with {b, y, z}, and {c, d, e, f, g, x} outside both circles in U.

(e) D = {a, b} and E = {b, a}

Solution:

D = {a, b}

E = {b, a}

For the Venn diagram:

- D ∩ E = {a, b} (common elements).

- D = E (D and E are equal).

- Elements not in D or E: {c, d, e, f, g, x, y, z}.

The Venn diagram shows one circle (since D = E): D and E with {a, b}, and {c, d, e, f, g, x, y, z} outside the circle in U.

(f) B = {x, y, m} and S = {x, y, z}

Solution:

B = {x, y, m}

S = {x, y, z}

For the Venn diagram:

- B ∩ S = {x, y} (common elements).

- B - S = {m}, S - B = {z}.

- Elements not in B or S: {a, b, c, d, e, f, g}.

The Venn diagram shows two overlapping circles: B with {x, y, m} and S with {x, y, z}, overlapping at {x, y}, and {a, b, c, d, e, f, g} outside both circles in U.

Question 5: (a) If A = {4, 5, 6} and B = {1, 2, 4}, show it in a Venn diagram. (b) If P = {2, 4, 6, 8, 10} and Q = {even numbers less than 10}, show sets P and Q in a Venn diagram. (c) If A = {a, b, c} and B = {1, 2, 3}, show sets A and B in a Venn diagram.

Solution:

(a) A = {4, 5, 6} and B = {1, 2, 4}

Solution:

A = {4, 5, 6}

B = {1, 2, 4}

Universal set U is the union of all elements (assuming a reasonable context, e.g., numbers 1 to 10):

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

For the Venn diagram:

- A ∩ B = {4} (common element).

- A - B = {5, 6}, B - A = {1, 2}.

- Elements not in A or B (within U): {3, 7, 8, 9, 10}.

The Venn diagram shows two overlapping circles: A with {4, 5, 6} and B with {1, 2, 4}, overlapping at {4}, and {3, 7, 8, 9, 10} outside both circles in U.

(b) P = {2, 4, 6, 8, 10} and Q = {even numbers less than 10}

Solution:

P = {2, 4, 6, 8, 10}

Q = {2, 4, 6, 8} (even numbers less than 10)

Universal set U is the union of all elements (assuming a reasonable context, e.g., numbers 1 to 10):

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

For the Venn diagram:

- Q ⊆ P (Q is a subset of P).

- P ∩ Q = {2, 4, 6, 8}.

- P - Q = {10}, Q - P = {}.

- Elements not in P or Q (within U): {1, 3, 5, 7, 9}.

The Venn diagram shows two circles: P with {2, 4, 6, 8, 10} and Q inside P with {2, 4, 6, 8}, and {1, 3, 5, 7, 9} outside both circles in U.

(c) A = {a, b, c} and B = {1, 2, 3}

Solution:

A = {a, b, c}

B = {1, 2, 3}

Universal set U (union of A and B):

U = {a, b, c, 1, 2, 3}

For the Venn diagram:

- A ∩ B = {} (no common elements).

- Elements not in A or B: {}.

The Venn diagram shows two separate circles: A with {a, b, c} and B with {1, 2, 3}, with no overlap, and no elements outside both circles in U.

Question 6: Draw Venn diagrams for the following pairs of sets and shade the overlapping parts: (a) X = {a, b, c, d} and Y = {b, c, e, f} (b) A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12, 15, 18} (c) R = {x : x is a factor of 12} and M = {x : x is a factor of 16} (d) S = {letters needed to spell the word 'donkey'} and D = {letters needed to spell the word 'monkey'} (e) N = {mountain, terai, himalaya} and E = {Bagamati, Narayan, Gandaki}

Solution:

(a) X = {a, b, c, d} and Y = {b, c, e, f}

Solution:

X = {a, b, c, d}

Y = {b, c, e, f}

Universal set U (union of X and Y): {a, b, c, d, e, f}

For the Venn diagram:

- X ∩ Y = {b, c} (common elements).

- X - Y = {a, d}, Y - X = {e, f}.

The Venn diagram shows two overlapping circles: X with {a, b, c, d} and Y with {b, c, e, f}, overlapping at {b, c} (shade {b, c}).

(b) A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12, 15, 18}

Solution:

A = {2, 4, 6, 8, 10, 12}

B = {3, 6, 9, 12, 15, 18}

Universal set U (union of A and B, assuming a reasonable context, e.g., numbers 1 to 18): {1, 2, 3, ..., 18}

For the Venn diagram:

A ∩ B = {6, 12} (common elements).

A - B = {2, 4, 8, 10}, B - A = {3, 9, 15, 18}.

Elements not in A or B (within U): {1, 5, 7, 11, 13, 14, 16, 17}.

The Venn diagram shows two overlapping circles: A with {2, 4, 6, 8, 10, 12} and B with {3, 6, 9, 12, 15, 18}, overlapping at {6, 12} (shade {6, 12}).

(c) R = {x : x is a factor of 12} and M = {x : x is a factor of 16}

Solution:

R = {1, 2, 3, 4, 6, 12} (factors of 12)

M = {1, 2, 4, 8, 16} (factors of 16)

Universal set U (union of R and M, assuming a reasonable context, e.g., numbers 1 to 16): {1, 2, 3, ..., 16}

For the Venn diagram:

R ∩ M = {1, 2, 4} (common elements).

R - M = {3, 6, 12}, M - R = {8, 16}.

Elements not in R or M (within U): {5, 7, 9, 10, 11, 13, 14, 15}.

The Venn diagram shows two overlapping circles: R with {1, 2, 3, 4, 6, 12} and M with {1, 2, 4, 8, 16}, overlapping at {1, 2, 4} (shade {1, 2, 4}).

(d) S = {letters needed to spell the word 'donkey'} and D = {letters needed to spell the word 'monkey'}

Solution:

S = {d, o, n, k, e, y} (letters in "donkey")

D = {m, o, n, k, e, y} (letters in "monkey")

Universal set U (union of S and D): {d, o, n, k, e, y, m}

For the Venn diagram:

S ∩ D = {o, n, k, e, y} (common elements).

S - D = {d}, D - S = {m}.

The Venn diagram shows two overlapping circles: S with {d, o, n, k, e, y} and D with {m, o, n, k, e, y}, overlapping at {o, n, k, e, y} (shade {o, n, k, e, y}).

(e) N = {mountain, terai, himalaya} and E = {Bagamati, Narayan, Gandaki}

Solution:

N = {mountain, terai, himalaya}

E = {Bagamati, Narayan, Gandaki}

Universal set U (union of N and E): {mountain, terai, himalaya, Bagamati, Narayan, Gandaki}

For the Venn diagram:

N ∩ E = {} (no common elements).

The Venn diagram shows two separate circles: N with {mountain, terai, himalaya} and E with {Bagamati, Narayan, Gandaki}, with no overlap (nothing to shade).

Question 7: Study the given Venn diagram and: (a) List the elements of set U (b) List the elements of set A (c) List the elements of set B

Solution:

(a) List the elements of set U

Solution:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(b) List the elements of set A

Solution:

A = {1, 2, 3, 4, 5, 6}

(c) List the elements of set B

Solution:

B = {2,4, 6, 7, 8, 9}

Question 8: Study the given Venn diagram and: (a) List the elements of set U (b) List the elements of set R (c) List the elements of set N

Solution:

U = {a, b, c, d, e, f, g, h, i, j}

(b) List the elements of set R

Solution:

R = {a, b, c, d, e, f}

(c) List the elements of set N

Solution:

N = {d, e, f, g, h, i}

EXERCISE 1. 2

Question 1: Define the following: (a) Subset (b) Superset (c) Power set (d) Proper subset (e) Improper subset

(a) Subset

A set A is a subset of set B if every element of A is also an element of B. This is denoted as A ⊆ B.

(b) Superset

A set B is a superset of set A if every element of A is an element of B. This is denoted as B ⊇ A.

(c) Power set

The power set of a set A, denoted P(A), is the set of all possible subsets of A, including the empty set and A itself.

(d) Proper subset

A set A is a proper subset of set B if A is a subset of B and A is not equal to B. This is denoted as A ⊂ B.

(e) Improper subset

A set A is an improper subset of set B if A is equal to B. An improper subset is a subset that includes all elements of the set itself.

Question 2: If A = {a, b, c} and B = {a, b, c, d, e}, then which set is a subset of the other?

Solution:

A = {a, b, c}

B = {a, b, c, d, e}

A set A is a subset of B if all elements of A are in B.

All elements of A (a, b, c) are in B (a, b, c, d, e).

Not all elements of B are in A (e.g., d, e are not in A).

Therefore, A is a subset of B.

Question 3: If A = {1, 2, 3}, B = {4, 5, 6}, and C = {1, 2}, which set is a subset of A?

Solution:

A = {1, 2, 3}

B = {4, 5, 6}

C = {1, 2}

A set is a subset of A if all its elements are in A.

B has elements (4, 5, 6), which are not in A (1, 2, 3), so B is not a subset of A.

C has elements (1, 2), which are in A (1, 2, 3), so C is a subset of A.

Therefore, C is a subset of A.

Question 4: If X = {10, 20, 30, 40} and Y = {20, 30}, determine whether Y is a proper subset of X.

Solution:

X = {10, 20, 30, 40}

Y = {20, 30}

A set Y is a proper subset of X if all elements of Y are in X and Y is not equal to X.

Elements of Y (20, 30) are in X (10, 20, 30, 40).

Y is not equal to X (Y has 2 elements, X has 4 elements).

Therefore, Y is a proper subset of X.

Question 5: If X = {m, a, i, n}, Y = {a, i, m}, and Z = {n, i, m, a}, which is a proper subset of X?

Solution:

X = {m, a, i, n}

Y = {a, i, m}

Z = {n, i, m, a}

A set is a proper subset of X if all its elements are in X and it is not equal to X.

All elements of Y (a, i, m) are in X, and Y is not equal to X (Y has 3 elements, X has 4 elements).

All elements of Z (n, i, m, a) are in X, but Z is equal to X (same elements).

Therefore, Y is a proper subset of X.

Question 6: If T = {2, 4, 6}, O = {1, 3, 4}, and M = {4, 6, 2}, which is an improper subset of T?

Solution:

T = {2, 4, 6}

O = {1, 3, 4}

M = {4, 6, 2}

An improper subset of T is a set that is equal to T.

O has element 1, which is not in T, so O is not a subset of T.

M = {4, 6, 2} has the same elements as T = {2, 4, 6}, so M is equal to T.

Therefore, M is an improper subset of T.

Question 7: If P = {a, b}, list all its proper subsets.

Solution:

P = {a, b}

The proper subsets are {}, {a}, {b}.

Question 8: If D = {3, 6, 9}, list all its proper subsets.

Solution:

D = {3, 6, 9}

The proper subsets are {}, {3}, {6}, {9}, {3, 6}, {3, 9}, {6, 9}.

Question 9: What is the correct mathematical notation for saying "M is a proper subset of N"?

Solution:

The notation for "M is a proper subset of N" is M ⊂ N, where M is a subset of N and M is not equal to N.

Question 10: Based on the given Venn diagram, which set is a proper subset of set I?

Solution:

The Venn diagram shows sets A and I within the universal set U, with M contained entirely within I.

Since M is entirely within I, M ⊆ M, and M ≠ I (M is smaller than I).

Therefore, M is a proper subset of I.

Question 11: What is the correct formula to calculate number of subsets?

Solution:

If a set has n elements, the number of subsets is 2ⁿ.

Therefore, the formula is 2ⁿ, where n is the number of elements in the set.

Question 12: What is the correct formula to calculate number of proper subsets?

Solution:

If a set has n elements, the number of subsets is 2ⁿ.

The number of proper subsets is 2ⁿ - 1 (excluding the set itself).

Therefore, the formula is 2ⁿ - 1, where n is the number of elements in the set.

Question 13: If a set has 3 elements, how many subsets does it have?

Solution:

Number of elements n = 3

Number of subsets = 2ⁿ

= 2³

= 8

Therefore, the set has 8 subsets.

SOLVE 

Question 1: If A = {a, b, c, d, e, f}, B = {a, b, c}, and C = {a, b, c, d}, then find the following:

Solution:

(a) n(A)

A = {a, b, c, d, e, f}

n(A) = 6

(b) n(B)

B = {a, b, c}

n(B) = 3

(c) n(C)

C = {a, b, c, d}

n(C) = 4

(d) Number of subsets of A

Number of subsets of a set with n elements = 2ⁿ

n(A) = 6

Number of subsets of A = 2⁶

= 64

(e) Number of subsets of B

n(B) = 3

Number of subsets of B = 2³

= 8

(f) Number of proper subsets of C

Number of proper subsets = 2ⁿ

n(C) = 4

Number of proper subsets of C = 2⁴

= 16

(g) Number of proper subsets of B

B = {a, b, c}

n(B) = 3

Number of proper subsets of C = 2³ - 1

= 8 - 1

= 7

(h) Number of proper subsets of A

Number of proper subsets = 2ⁿ - 1

n(A) = 6

Number of proper subsets of A = 2⁶ - 1

= 64 - 1

= 63

Question 2: Using the data from question number 1, write down the relation between the following sets:

Solution:

(a) B ⊂ A

(b) B ⊂ C

(c) C ⊂ A

(d) A ⊂ B

(e) {a, b} ⊂ C

(f) {a, b, c, d} ⊂ A

Question 3: The set of students of a class is given below: A = {Raju, Jyoti, Rakshya, Sunam, Januka, Suman}. List the elements of the following subsets of A.

Solution:

(a) Whose name start with R

= {Raju, Rakshya}

(b) Whose name start with J

= {Jyoti, Januka}

(c) Whose name start with S

= {Sunam, Suman}

(d) Whose name start with K

= {}

(e) Whose name start with R or S

Names starting with R = {Raju, Rakshya}

Names starting with S = {Sunam, Suman}

R or S = {Raju, Rakshya, Sunam, Suman}

(f) Whose name start with S or J

Names starting with S = {Sunam, Suman}

Names starting with J = {Jyoti, Januka}

S or J = {Sunam, Suman, Jyoti, Januka}

(g) Whose name start with R or J

Names starting with R = {Raju, Rakshya}

Names starting with J = {Jyoti, Januka}

R or J = {Raju, Rakshya, Jyoti, Januka}

Question 3: The set of students of a class is given below: A = {Raju, Jyoti, Rakshya, Sunam, Januka, Suman}. List the elements of the following subsets of A.

Solution:

(h) Whose names start with R, J, or S

Names starting with J = {Jyoti, Januka}

Names starting with R = {Raju, Rakshya}

Names starting with S = {Sunam, Suman}

R, J, or S = {Jyoti, Januka, Raju, Rakshya, Sunam, Suman}

Question 4: Let A = {2, 4, 6, 8, 10, 12} be the given set. List the following subsets of A:

Solution:

(a) Q = Set of odd numbers

Q = {}

(b) P = Set of prime numbers

P = {2}

(c) C = Set of composite numbers

= {4, 6, 8, 10, 12}

(d) F₁₂ = Set of factors of 12

Factors of 12 = {1, 2, 3, 4, 6, 12}

Factors in A = {2, 4, 6, 12}

= {2, 4, 6, 12}

(e) M₂ = Set of multiples of 2

= {2, 4, 6, 8, 10, 12}

(f) M₁₅ = Set of multiples of 15

= {}

5. (a) Write all the proper subsets of the set W = {w, x, y}.

Solution:

W = {w, x, y}

n(W) = 3

Number of proper subsets = 2^3 - 1 = 8 - 1 = 7

Proper subsets are:

{}, {w}, {x}, {y}, {w, x}, {w, y}, {x, y}

(b) Write all the proper subsets of the set P = {13, 17, 19}.

Solution:

P = {13, 17, 19}

n(P) = 3

Number of proper subsets = 2^3 - 1 = 8 - 1 = 7

Proper subsets are:

{}, {13}, {17}, {19}, {13, 17}, {13, 19}, {17, 19}

6. Let, N = the set of all private school students, Z = the set of all private school students of Chitwan, Q = the set of all students of a private school in Chitwan, T = the set of all students of grade 7 of a private school in Chitwan. Then choose the true statement:

Solution:

(a) N ⊂ Z: False, because N includes all private school students, not just those in Chitwan.

(b) T ⊂ Q: True, because grade 7 students (T) of a private school in Chitwan are part of all students of that school (Q).

(c) Z ⊂ N: True, because all private school students in Chitwan (Z) are part of all private school students (N).

(d) Q ⊂ Z: True, because students of a specific private school in Chitwan (Q) are part of all private school students in Chitwan (Z).

Question 7: If A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}, and C = {1, 2, 3}, then which of the following statements are true?

Solution:

We evaluate each statement to determine which are true:

(a) A ⊂ B

A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}

Elements 1, 3, 5 in A are not in B.

Therefore, A ⊂ B is false.

(b) B ⊂ A

B = {2, 4, 6, 8}, A = {1, 2, 3, 4, 5, 6}

Element 8 in B is not in A.

Therefore, B ⊂ A is false.

(c) C ⊂ A

C = {1, 2, 3}, A = {1, 2, 3, 4, 5, 6}

All elements of C (1, 2, 3) are in A, and C ≠ A.

Therefore, C ⊂ A is true.

(d) B ⊃ C (interpreted as C ⊂ B)

B = {2, 4, 6, 8}, C = {1, 2, 3}

Elements 1, 3 in C are not in B.

Therefore, C ⊂ B is false.

(e) \( A \not\subset B \)

A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}

Elements 1, 3, 5 in A are not in B, so A is not a subset of B.

Therefore, A \not⊂ B is true.

(e) \( B \not\subset A \)

B = {2, 4, 6, 8}, A = {1, 2, 3, 4, 5, 6}

Element 8 in B is not in A, so B is not a subset of A.

Therefore, B \not⊂ A is true.

(g) C = B

C = {1, 2, 3}, B = {2, 4, 6, 8}

C and B have different elements (e.g., C has 1, 3; B has 4, 6, 8).

Therefore, C = B is false.

(h) A = B

A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}

A and B have different elements (e.g., A has 1, 3, 5; B has 8).

Therefore, A = B is false.

Question 8: If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, C = {6, 7, 8}, then which of the following statements are true?

Solution:

(a) A = B

A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}

A and B have different elements (e.g., A has 1, 2; B has 6, 7).

Therefore, A = B is false.

(b) A ⊂ B

A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}

Elements 1, 2 in A are not in B.

Therefore, A ⊂ B is false.

(c) B ⊂ U

B = {3, 4, 5, 6, 7}, U = {1, 2, 3, 4, 5, 6, 7, 8}

All elements of B (3, 4, 5, 6, 7) are in U, and B ≠ U.

Therefore, B ⊂ U is true.

9. If P = Set of all quadrilaterals, Q = Set of all squares, R = Set of all parallelograms, S = Set of all rectangles, then state which of the following statements are true.

Solution:

(a) Q ⊂ S: True, because all squares are rectangles.

(b) P ⊂ Q: False, because not all quadrilaterals are squares.

(c) S ⊂ R: True, because all rectangles are parallelograms.

(d) S ⊂ R ⊂ P: True, because S ⊂ R (rectangles are parallelograms) and R ⊂ P (parallelograms are quadrilaterals).

(e) Q ⊂ S ⊂ R ⊂ P: True, because Q ⊂ S, S ⊂ R, and R ⊂ P are all true.

(f) P = Q: False, because not all quadrilaterals are squares.

Question 10: If P = Set of all triangles, Q = Set of scalene triangles, R = Set of isosceles triangles, S = Set of equilateral triangles, then state which of the following statements are true or false.

Solution:

(a) Q ⊂ P

All scalene triangles are triangles.

Therefore, Q ⊂ P is true.

(b) R ⊂ S

Not all isosceles triangles are equilateral (e.g., a triangle with sides 2, 2, 3 is isosceles but not equilateral).

Therefore, R ⊂ S is false.

(c) S ⊃ Q (interpreted as Q ⊂ S)

Equilateral triangles have all sides equal, while scalene triangles have all sides different, so no scalene triangle is equilateral.

Therefore, Q ⊂ S is false.

(d) P ⊃ R (interpreted as R ⊂ P)

All isosceles triangles are triangles.

Therefore, R ⊂ P is true.

(e) S ⊂ R

All equilateral triangles have three equal sides, thus at least two equal sides, making them isosceles.

Therefore, S ⊂ R is true.

(f) P ⊂ Q

Not all triangles are scalene (e.g., equilateral triangles have all sides equal).

Therefore, P ⊂ Q is false.

11. (a) How many subsets can be obtained from the set A = {m, a, t, h}?

Solution:

A = {m, a, t, h}

n(A) = 4

Number of subsets = 2ⁿ

= 2⁴

= 16

Question (b): How many subsets can be obtained from the set B = {p, a, b, s, o, n}?

Solution:

B = {p, a, b, s, o, n}

n(B) = 6

Number of subsets = 2ⁿ

= 2⁶

= 64

Question 12: Write the number of subsets of the following sets:

Solution:

(a) A = {1}

n(A) = 1

Number of subsets = 2¹ = 2

(b) A = {1, 2}

n(A) = 2

Number of subsets = 2² = 4

(c) A = {1, 2, 3}

n(A) = 3

Number of subsets = 2³ = 8

(d) B = {b, c, a}

n(B) = 3

Number of subsets = 2³ = 8

(e) B = {g, o, i, d}

n(B) = 4

Number of subsets = 2⁴ = 16

(f) B = {1, 2, 3, 4, 5}

n(B) = 5

Number of subsets = 2⁵ = 32

(g) C = {first three prime numbers}

First three primes = {2, 3, 5}

n(C) = 3

Number of subsets = 2³ = 8

(h) C = {first two square numbers}

First two squares = {1, 4}

n(C) = 2

Number of subsets = 2² = 4

(i) C = {first four cube numbers}

First four cubes = {1, 8, 27, 64}

n(C) = 4

Number of subsets = 2⁴ = 16

Question 13: Given that U = {2, 3, 5, 7, 11, 12, 17, 21, 25, 26, 35, 42}, A = {x : x ∈ U and x is an odd number}, B = {x : x ∈ U and x is a multiple of 7}, C = {x : x ∈ U and x is a prime number}. List the elements of the following sets:

Solution:

(a) A

A = {x : x ∈ U and x is an odd number}

Odd numbers in U = {3, 5, 7, 11, 17, 21, 25, 35}

Therefore, A = {3, 5, 7, 11, 17, 21, 25, 35}

(b) B

B = {x : x ∈ U and x is a multiple of 7}

Multiples of 7 in U = {7, 21, 35, 42}

Therefore, B = {7, 21, 35, 42}

(c) C

C = {x : x ∈ U and x is a prime number}

Prime numbers in U = {2, 3, 5, 7, 11, 17}

Therefore, C = {2, 3, 5, 7, 11, 17}

(d) Set of elements belonging to both B and C

B = {7, 21, 35, 42}, C = {2, 3, 5, 7, 11, 17}

Common elements = {7}

Therefore, B ∩ C = {7}

Question 14: Let U = {10, 11, 12, ..., 20}. List the following sets:

Solution:

(a) A = {x : x is an even number}

Even numbers in U = {10, 12, 14, 16, 18, 20}

Therefore, A = {10, 12, 14, 16, 18, 20}

(b) X = {x : x is an odd number}

Odd numbers in U = {11, 13, 15, 17, 19}

Therefore, X = {11, 13, 15, 17, 19}

(c) Z = {x : x is a square number}

Square numbers in U = {16}

Therefore, Z = {16}

(d) Y = {x : x is a prime number}

Prime numbers in U = {11, 13, 17, 19}

Therefore, Y = {11, 13, 17, 19}

15. If U = {x : x ∈ ℕ, x ≤ 10} is a universal set, A = {y : y is an even number, y ∈ U}, and B = {z : z is a prime number, z ∈ U}, then:

Solution:

(a) List the elements of U:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(b) List the elements of A:

A = {y : y is an even number, y ∈ U}

A = {2, 4, 6, 8, 10}

(c) List the elements of B:

B = {z : z is a prime number, z ∈ U}

B = {2, 3, 5, 7}

(d) Is A ⊂ B?

A = {2, 4, 6, 8, 10}, B = {2, 3, 5, 7}

Elements 4, 6, 8, 10 in A are not in B.

So, A is not a subset of B.

(e) Is B ⊂ A?

Elements 3, 5, 7 in B are not in A.

So, B is not a subset of A.

(f) Are the sets A and B equal?

A and B have different elements.

So, A and B are not equal.

Question 16: If U = {x : x ∈ ℕ, x ≤ 12} is a universal set, A = {y : y is an odd number, y ∈ U}, and B = {z : z is an even number, z ∈ U}, answer the following:

Solution:

(a) List the elements of U

U = {x : x ∈ ℕ, x ≤ 12}

Natural numbers from 1 to 12: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Therefore, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

(b) List the elements of A

A = {y : y is an odd number, y ∈ U}

Odd numbers in U = {1, 3, 5, 7, 9, 11}

Therefore, A = {1, 3, 5, 7, 9, 11}

(c) List the elements of B

B = {z : z is an even number, z ∈ U}

Even numbers in U = {2, 4, 6, 8, 10, 12}

Therefore, B = {2, 4, 6, 8, 10, 12}

(d) Is \( A \subseteq B \)?

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

For A ⊂ B, all elements of A must be in B. Elements 1, 3, 5, 7, 9, 11 are not in B.

Therefore, A ⊂ B is false.

(e) Is \( B \subseteq A \)?

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

For B ⊂ A, all elements of B must be in A. Elements 2, 4, 6, 8, 10, 12 are not in A.

Therefore, B ⊂ A is false.

(f) Are the sets A and B equal?

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

A and B have no common elements, so they are not equal.

Therefore, A = B is false.