Sets — Exercise 5(B)

Write down the following statement, using set notation:

Set A is a proper subset of set B.

Answer

If every element of A is in B and there exists at least one element in B which is not there in A, then A is called a proper subset of B and we write A ⊂ B.

Hence, set A is a proper subset of set B is written as A ⊂ B.

Write down the following statement, using set notation:

Set C is a superset of set D.

Answer

If D is a subset of C, then we say that C is a superset of D and we write C ⊇ D.

Hence, set C is a superset of set D is written as C ⊇ D.

Write down the following statement, using set notation:

Set B contains set A.

Answer

If set B contains set A, then every element of A is in B, i.e., B is a superset of A and we write B ⊇ A.

Hence, set B contains set A is written as B ⊇ A.

Write down the following statement, using set notation:

Neither A is a subset of B, nor B is a subset of A.

Answer

If A is not a subset of B, we write A ⊄ B and if B is not a subset of A, we write B ⊄ A.

Hence, neither A is a subset of B, nor B is a subset of A is written as A ⊄ B and B ⊄ A.

Let A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses} in a plane. State, giving reasons, whether the following statement is true or false:

B ⊂ C ⊂ A

Answer

A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses}

B ⊂ C means every rectangle is a square.

A rectangle whose adjacent sides are unequal, for example a rectangle of length 5 cm and breadth 3 cm, is not a square.

So, there is a rectangle which is not a square.

∴ B ⊄ C

Hence, the given statement is false, as every rectangle is not a square.

Let A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses} in a plane. State, giving reasons, whether the following statement is true or false:

C ⊂ B ⊂ A

Answer

A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses}

Every square is a rectangle whose adjacent sides are equal, but every rectangle is not a square.

∴ C ⊂ B

Every rectangle is a quadrilateral, but every quadrilateral is not a rectangle.

∴ B ⊂ A

∴ C ⊂ B ⊂ A

Hence, the given statement is true, as every square is a rectangle and every rectangle is a quadrilateral.

Let A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses} in a plane. State, giving reasons, whether the following statement is true or false:

C ⊂ D ⊂ A

Answer

A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses}

Every square is a rhombus whose angles are right angles, but every rhombus is not a square.

∴ C ⊂ D

Every rhombus is a quadrilateral, but every quadrilateral is not a rhombus.

∴ D ⊂ A

∴ C ⊂ D ⊂ A

Hence, the given statement is true, as every square is a rhombus and every rhombus is a quadrilateral.

Let A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses} in a plane. State, giving reasons, whether the following statement is true or false:

D ⊂ C ⊂ A

Answer

A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses}

D ⊂ C means every rhombus is a square.

A rhombus whose angles are not right angles is not a square.

So, there is a rhombus which is not a square.

∴ D ⊄ C

Hence, the given statement is false, as every rhombus is not a square.

Let A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses} in a plane. State, giving reasons, whether the following statement is true or false:

A ⊇ B ⊇ C

Answer

A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses}

Every rectangle is a quadrilateral, so B is a subset of A.

∴ A ⊇ B

Every square is a rectangle, so C is a subset of B.

∴ B ⊇ C

∴ A ⊇ B ⊇ C

Hence, the given statement is true, as the set of quadrilaterals contains the set of rectangles which contains the set of squares.

Let A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses} in a plane. State, giving reasons, whether the following statement is true or false:

A ⊆ B ⊆ C

Answer

A = {all quadrilaterals}, B = {all rectangles}, C = {all squares} and D = {all rhombuses}

A ⊆ B means every quadrilateral is a rectangle.

A quadrilateral whose angles are not right angles, for example a trapezium, is not a rectangle.

∴ A ⊄ B

Hence, the given statement is false, as every quadrilateral is not a rectangle.

Let A = {all triangles}, B = {all isosceles triangles} and C = {all equilateral triangles}. State, giving reasons, whether the following statement is true or false:

B ⊂ C ⊂ A

Answer

A = {all triangles}, B = {all isosceles triangles} and C = {all equilateral triangles}

B ⊂ C means every isosceles triangle is an equilateral triangle.

An isosceles triangle whose sides are 5 cm, 5 cm and 3 cm does not have all its sides equal, so it is not an equilateral triangle.

∴ B ⊄ C

Hence, the given statement is false, as every isosceles triangle is not an equilateral triangle.

Let A = {all triangles}, B = {all isosceles triangles} and C = {all equilateral triangles}. State, giving reasons, whether the following statement is true or false:

C ⊂ B ⊂ A

Answer

A = {all triangles}, B = {all isosceles triangles} and C = {all equilateral triangles}

An equilateral triangle has all its three sides equal, so it has at least two sides equal and hence it is an isosceles triangle. But every isosceles triangle is not an equilateral triangle.

∴ C ⊂ B

Every isosceles triangle is a triangle, but every triangle is not an isosceles triangle.

∴ B ⊂ A

∴ C ⊂ B ⊂ A

Hence, the given statement is true, as every equilateral triangle is an isosceles triangle and every isosceles triangle is a triangle.

Let A = {1, 2}. State whether the following statement is true:

1 ⊂ A

Answer

A = {1, 2}

The symbol ⊂ is used between two sets.

Here 1 is an element of A and not a set, so we cannot write 1 ⊂ A.

Hence, the given statement is not true.

Let A = {1, 2}. State whether the following statement is true:

{1} ∈ A

Answer

A = {1, 2}

The elements of A are 1 and 2.

As {1} is a set and it is not an element of A,

∴ {1} ∉ A

Hence, the given statement is not true.

Let A = {1, 2}. State whether the following statement is true:

1 ∈ A

Answer

A = {1, 2}

The elements of A are 1 and 2.

As 1 is an element of A,

∴ 1 ∈ A

Hence, the given statement is true.

Let A = {1, 2}. State whether the following statement is true:

φ ∈ A

Answer

A = {1, 2}

The elements of A are 1 and 2.

As φ is not an element of A,

∴ φ ∉ A

Hence, the given statement is not true.

Let A = {1, 2}. State whether the following statement is true:

φ ⊂ A

Answer

A = {1, 2}

The empty set is a subset of every set.

Also, A is a non-empty set, so φ ≠ A.

∴ φ ⊂ A

Hence, the given statement is true.

Let A = {1, 2}. State whether the following statement is true:

{1} ∉ A

Answer

A = {1, 2}

The elements of A are 1 and 2.

As {1} is a set and it is not an element of A,

∴ {1} ∉ A

Hence, the given statement is true.

Is the following statement correct?

a ⊂ {a, b, c}

Answer

The symbol ⊂ is used between two sets. Here a is an element of {a, b, c} and not a set, so we cannot write a ⊂ {a, b, c}.

Hence, the given statement is not correct.

Is the following statement correct?

{a} ∈ {a, b, c}

Answer

The elements of {a, b, c} are a, b and c. As {a} is a set and it is not an element of {a, b, c},

∴ {a} ∉ {a, b, c}

Hence, the given statement is not correct.

Is the following statement correct?

{a} ⊂ {a, b, c}

Answer

If every element of a set P is in a set Q and there exists at least one element in Q which is not there in P, then P ⊂ Q. The only element of {a} is a and a ∈ {a, b, c}. Also, b ∈ {a, b, c} but b ∉ {a}.

∴ {a} ⊂ {a, b, c}

Hence, the given statement is correct.

Is the following statement correct?

φ ∈ {a, b, c}

Answer

The elements of {a, b, c} are a, b and c. As φ is not an element of {a, b, c},

∴ φ ∉ {a, b, c}

Hence, the given statement is not correct.

Is the following statement correct?

φ ⊂ {a, b, c}

Answer

The empty set is a subset of every set. Also, {a, b, c} is a non-empty set, so φ ≠ {a, b, c}.

∴ φ ⊂ {a, b, c}

Hence, the given statement is correct.

Is the following statement correct?

{φ} ⊂ {a, b, c}

Answer

The only element of {φ} is φ. As φ ∉ {a, b, c}, every element of {φ} is not in {a, b, c}.

∴ {φ} ⊄ {a, b, c}

Hence, the given statement is not correct.

Is the following statement correct?

a ∈ {{a}, b}

Answer

The elements of {{a}, b} are {a} and b. As a is not an element of {{a}, b},

∴ a ∉ {{a}, b}

Hence, the given statement is not correct.

Is the following statement correct?

{a} ⊆ {{a}, b}

Answer

The only element of {a} is a and the elements of {{a}, b} are {a} and b. As a ∉ {{a}, b}, every element of {a} is not in {{a}, b}.

∴ {a} ⊄ {{a}, b}

Hence, the given statement is not correct.

Is the following statement correct?

{a, b} ∈ {{a, b}, c}

Answer

The elements of {{a, b}, c} are {a, b} and c. As {a, b} is an element of {{a, b}, c},

∴ {a, b} ∈ {{a, b}, c}

Hence, the given statement is correct.

Is the following statement true?

φ = {0}

Answer

φ is a set containing no element at all. {0} is a set containing one element, namely 0. As 0 ∈ {0} but 0 ∉ φ,

∴ φ ≠ {0}

Hence, the given statement is not true.

Is the following statement true?

φ = {φ}

Answer

φ is a set containing no element at all. {φ} is a set containing one element, namely φ. As φ ∈ {φ} but φ has no element at all,

∴ φ ≠ {φ}

Hence, the given statement is not true.

Is the following statement true?

φ ∈ {0}

Answer

The only element of {0} is 0. As φ is not an element of {0},

∴ φ ∉ {0}

Hence, the given statement is not true.

Is the following statement true?

φ ∈ {φ}

Answer

The only element of {φ} is φ. As φ is an element of {φ},

∴ φ ∈ {φ}

Hence, the given statement is true.

Is the following statement true?

φ ∈ {φ, {0}}

Answer

The elements of {φ, {0}} are φ and {0}. As φ is an element of {φ, {0}},

∴ φ ∈ {φ, {0}}

Hence, the given statement is true.

Is the following statement true?

{φ} ⊂ {0}

Answer

The only element of {φ} is φ and the only element of {0} is 0. As φ ∉ {0}, every element of {φ} is not in {0}.

∴ {φ} ⊄ {0}

Hence, the given statement is not true.

Is the following statement true?

For any two sets A and B, either A ⊆ B or B ⊆ A.

Answer

Let A = {1, 2} and B = {2, 3}.

Here 1 ∈ A but 1 ∉ B, so A ⊄ B.

Also, 3 ∈ B but 3 ∉ A, so B ⊄ A.

So, there are two sets A and B such that neither A ⊆ B nor B ⊆ A.

Hence, the given statement is not true.

Is the following statement true?

Every subset of a finite set is finite.

Answer

A finite set has a countable number of elements.

Every element of a subset of a finite set is also an element of that finite set.

So, a subset of a finite set cannot have more elements than the finite set itself and hence the process of counting its elements surely comes to an end.

Hence, the given statement is true.

Is the following statement true?

Every subset of an infinite set is infinite.

Answer

Let N = {1, 2, 3, 4, 5, ....} be the set of all natural numbers, which is an infinite set.

Let A = {1, 2, 3}.

Every element of A is in N, so A is a subset of N.

But A has only 3 elements, so A is a finite set.

So, there is a subset of an infinite set which is finite.

Hence, the given statement is not true.

Is the following statement true?

Every set has a proper subset.

Answer

If A is a proper subset of B, then every element of A is in B and there exists at least one element in B which is not there in A.

The empty set φ has no element at all, so there can be no set which has fewer elements than φ.

∴ φ has no proper subset.

Hence, the given statement is not true.

Is the following statement true?

If A has n elements, then P(A) has 2ⁿ subsets.

Answer

The set of all possible subsets of a set A is called the power set of A, denoted by P(A).

If A contains n elements, then A has 2ⁿ subsets and all of these subsets are the members of P(A).

So, P(A) consists of the 2ⁿ subsets of A.

Hence, the given statement is true.

Let A be the set of letters in the word, 'seed'. Find

(i) A

(ii) n(A)

(iii) number of subsets of A.

(iv) number of proper subsets of A.

Answer

(i) A is the set of letters in the word 'seed'.

The letters of the word 'seed' are s, e, e, d.

As every repeated element is taken only once in a set, writing each letter once and only once, we get

Hence, A = {s, e, d}.

(ii) A = {s, e, d}

As A has 3 elements,

Hence, n(A) = 3.

(iii) If a set contains n elements, then it has 2ⁿ subsets.

Here n(A) = 3.

∴ Number of subsets of A = 2³ = 8

Hence, the number of subsets of A is 8.

(iv) If a set contains n elements, then it has (2ⁿ - 1) proper subsets.

Here n(A) = 3.

∴ Number of proper subsets of A = 2³ - 1 = 8 - 1 = 7

Hence, the number of proper subsets of A is 7.

Find all possible subsets of the following set:

A = {4, 9}

Answer

A = {4, 9}

If a set contains n elements, then it has 2ⁿ subsets.

As A contains 2 elements, there are 2² = 4 subsets of A.

The empty set is a subset of every set and every set is a subset of itself.

So, all possible subsets of A are A₁ = φ, A₂ = {4}, A₃ = {9} and A₄ = {4, 9}.

Hence, all possible subsets of A are φ, {4}, {9} and {4, 9}.

Find all possible subsets of the following set:

B = {2, 3, 8}

Answer

B = {2, 3, 8}

If a set contains n elements, then it has 2ⁿ subsets.

As B contains 3 elements, there are 2³ = 8 subsets of B.

The empty set is a subset of every set and every set is a subset of itself.

So, all possible subsets of B are B₁ = φ, B₂ = {2}, B₃ = {3}, B₄ = {8}, B₅ = {2, 3}, B₆ = {3, 8}, B₇ = {2, 8} and B₈ = {2, 3, 8}.

Hence, all possible subsets of B are φ, {2}, {3}, {8}, {2, 3}, {3, 8}, {2, 8} and {2, 3, 8}.

Find all possible subsets of the following set:

C = {0, 1, 2}

Answer

C = {0, 1, 2}

If a set contains n elements, then it has 2ⁿ subsets.

As C contains 3 elements, there are 2³ = 8 subsets of C.

The empty set is a subset of every set and every set is a subset of itself.

So, all possible subsets of C are C₁ = φ, C₂ = {0}, C₃ = {1}, C₄ = {2}, C₅ = {0, 1}, C₆ = {1, 2}, C₇ = {0, 2} and C₈ = {0, 1, 2}.

Hence, all possible subsets of C are φ, {0}, {1}, {2}, {0, 1}, {1, 2}, {0, 2} and {0, 1, 2}.

Find the power set of the following set:

A = {0, 5}

Answer

A = {0, 5}

The set of all possible subsets of a set A is called the power set of A, denoted by P(A).

As A contains 2 elements, there are 2² = 4 subsets of A.

All possible subsets of A are φ, {0}, {5} and {0, 5}.

∴ P(A) = Set of all possible subsets of A

Hence, P(A) = {φ, {0}, {5}, {0, 5}}.

Find the power set of the following set:

B = {7, 9}

Answer

B = {7, 9}

The set of all possible subsets of a set B is called the power set of B, denoted by P(B).

As B contains 2 elements, there are 2² = 4 subsets of B.

All possible subsets of B are φ, {7}, {9} and {7, 9}.

∴ P(B) = Set of all possible subsets of B

Hence, P(B) = {φ, {7}, {9}, {7, 9}}.

Find the power set of the following set:

C = {2, 4, 6}

Answer

C = {2, 4, 6}

The set of all possible subsets of a set C is called the power set of C, denoted by P(C).

As C contains 3 elements, there are 2³ = 8 subsets of C.

All possible subsets of C are φ, {2}, {4}, {6}, {2, 4}, {4, 6}, {2, 6} and {2, 4, 6}.

∴ P(C) = Set of all possible subsets of C

Hence, P(C) = {φ, {2}, {4}, {6}, {2, 4}, {4, 6}, {2, 6}, {2, 4, 6}}.

Let A = {1, {2}}. Find the power set of A.

Answer

A = {1, {2}}

Let A = {1, a}, where a = {2}.

The set of all possible subsets of a set A is called the power set of A, denoted by P(A).

As A contains 2 elements, there are 2² = 4 subsets of A.

All possible subsets of A are A₁ = φ, A₂ = {1}, A₃ = {a} and A₄ = {1, a}.

∴ P(A) = {φ, {1}, {a}, {1, a}}

Replacing a by {2}, we get

Hence, P(A) = {φ, {1}, {{2}}, {1, {2}}}.

Let ξ = {x : x ∈ N, x < 50}, A = {x : x² ∈ ξ}, B = {x : x = n², n ∈ N} and C = {x : x is a factor of 36}. List the elements of each of the sets A, B and C. Also, state whether each of the following statements is true or false:

(i) A ⊆ B

(ii) A = B

(iii) A ↔ B

(iv) B ↔ C

(v) n(A) < n(C)

Answer

ξ = {x : x ∈ N, x < 50}

ξ = {1, 2, 3, ...., 49}

A = {x : x² ∈ ξ}

Here x² ≤ 49, i.e., x ≤ 7.

As 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36 and 7² = 49 all belong to ξ, while 8² = 64 does not belong to ξ,

∴ A = {1, 2, 3, 4, 5, 6, 7}

B = {x : x = n², n ∈ N}

Putting n = 1, 2, 3, 4, 5, 6 and 7 in n², the squares which belong to ξ are 1, 4, 9, 16, 25, 36 and 49.

∴ B = {1, 4, 9, 16, 25, 36, 49}

C = {x : x is a factor of 36}

36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6

∴ C = {1, 2, 3, 4, 6, 9, 12, 18, 36}

Here n(A) = 7, n(B) = 7 and n(C) = 9.

(i) A ⊆ B

As 2 ∈ A but 2 ∉ B, every element of A is not in B.

∴ A ⊄ B

Hence, the given statement is false.

(ii) A = B

Two sets A and B are said to be equal, if every element of A is in B and every element of B is in A.

As 2 ∈ A but 2 ∉ B,

∴ A ≠ B

Hence, the given statement is false.

(iii) A ↔ B

Two finite sets A and B are said to be equivalent, if n(A) = n(B).

As n(A) = 7 and n(B) = 7, so n(A) = n(B).

∴ A ↔ B

Hence, the given statement is true.

(iv) B ↔ C

Two finite sets B and C are said to be equivalent, if n(B) = n(C).

As n(B) = 7 and n(C) = 9, so n(B) ≠ n(C).

∴ B and C are not equivalent sets.

Hence, the given statement is false.

(v) n(A) < n(C)

As n(A) = 7 and n(C) = 9 and 7 < 9,

∴ n(A) < n(C)

Hence, the given statement is true.