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```Chapter 1: Sets – Exercise 1.4

Sets – Exercise 1.4 – Q.1

(i) False, ∵ the two sets A and B need not be comparable.

(ii) False, ∵{1} is a finite subset of the infinite set N of natural numbers.

(iii) True, ∵ the order (or cardinal number) of any subset of a set is less than or equal to the order of the set. {order (or cardinal number) of a set is the number of elements in the set}.

(iv) False, ∵ the empty set ∮ has no proper subset.

(v) False, ∵ {a, b, a, b,...} = {a, b} (repetition is not allowed)

∴ {a, b, a, b,....} is a finite set.

(vi) True, ∵ equivalent sets have the same cardinal number.

(vii) False,

One knows that if the cardinal number of a set A is n, then the power set of A denoted by P(A) which is the set of all subsets of all, has the cardinal number 2n.

If the cardinal number of A is infinite, then the cardinal number of P (A) is also infinite. Hence, the above statement is true provided the set is infinite.

Sets – Exercise 1.4 – Q.2

(i) True, ∵ 1 is an element of the set {1, 2, 3}.

(ii) False, ∵ a is an element and not a subset of the set {b, c, a}.

(iii) False, ∵ {a} is a subset of the set {a, b, c} and not an element.

(iv) True, ∵ repetition is not allowed in a set.

(v) False, ∵ the set {x : x + 8 = 8} is the single ton set {0} which is not the null set ∮.

Sets – Exercise 1.4 – Q.3

We have,

A = {x : x satisfies x2 - 8x + 12 = 0}

= {x : x2 - 6x - 2x +12 = 0}

= {x : x (x - 6) - 2(x - 6) = 0}

= {x: (x - 6)(x - 2) = 0}

= {x : x = 6, 2}

= {6, 2}

B = {2, 4, 6}

C = {2, 4, 6, 8,....}

D = {6}

We know that if E and F are two sets, then E is a subset of F. i.e., E ⊆ F if x ϵ E ⟹ x ϵ F, E is called a proper subset of F if E is strictly contained in F and is denoted by E ⊂ F.

Clearly,

D ⊂ A {∵ 6 ϵ D and 6 ϵ A}

A ⊂ B{∵ 2,6 ϵ A and they also belong to B}

Similarly, B ⊂ C

Hence, D = A ⊂ B ⊂ C.

Sets – Exercise 1.4 – Q.4(i)

The given statement is 'True'.

If m ϵ z, then m can be written as m/1, which is of the form p/q  where p and q are relatively prime integers and q ≠ 0.

This implies that m ϵ Q, the set of rational numbers.

Thus, m ϵ z ⟹ m ϵ Q

Hence Z ⊆ Q

Sets – Exercise 1.4 – Q.4(ii)

The given statement is 'True'.

∵  Crows are also Birds.

Sets – Exercise 1.4 – Q.4(iii)

The given statement is 'False'.

A rectangle need not be a square.

Sets – Exercise 1.4 – Q.4(iv)

The given statement is 'True'

If z is a complex number, then it can be written as z = x + iy,

where x and y are real numbers and are called the real and imaginary parts of the complex number z.

If x is a real number,

then x = x + i, 0 ϵ C,

where C is the set of complex numbers.

Thus x ϵ R ⟹ x ϵ C

Hence, the set of all real numbers is contained in the set of all complex numbers.

Sets – Exercise 1.4 – Q.4(v)

False, ∵ a ϵ P but a ∉ B

Note that {a} is an element of B which is different from the element 'a'.

Sets – Exercise 1.4 – Q.4(vi)

(vi) A =  {L, I, T, E} [∵ repetition is not allowed]

B = {T,I,L,E}  [∵repetition is not allowed]

= {L, I, T, E}  [∵ the manner in which the elements are listed does not matter]

∵Each element of A is an element of B and vice-versa

∴ A = B

Hence, the given statement is true.

Sets – Exercise 1.4 – Q.5

(i) False,

The correct statement is a ϵ {a, b, c}.

(ii) False, ∵{a} is a subset and not an element of {a, b, c}

The correct form is {a} ⊂ {a, b, c}.

(iii) False, ∵ a is not an element of {{a}, b}

The correct form is {a} ϵ {{a},b}

(iv) False, ∵ {a} is not a subset of {{a},b} hence it cannot be contained in it.

The correct form is {a} ϵ {{a, b}. Another correct form could be {{a}} ⊂ {{a}, b}.

(v) False, ∵ {a, b,} is an element and not a subset of {a,{,bc}}.

The correct form is {b, c} ϵ (a,{b, c}}.

(vi) False, ∵ {a, b} is not a subset of {a, {b, c}}

The correct form is {a, b} ⊄ {a,{b, c}}.

(vii) False, ∵∮ is not an element of {a, b}.

The correct form is ∮ ⊂ {a, b}.

(viii) True, ∵ empty set φ is a subset of every set.

(ix) False, ∵ {x : x + 3 = 3} = {x : x = 0} = {0}

The correct form is {x : x + 3 = 3} ≠ ∮.

Sets – Exercise 1.4 – Q.6

(i) False, {c, d} is an element of A and not a subset of A.

(ii) True, ∵ {c, d} is indeed an element of A.

(iii) True, ∵ {c, d} is a subset of A.

(iv) True,

(v) False, ∵ a belongs to A and not a subset of A. An element of a set belongs to it whereas a subset of it is contained in it.

(vi) True, {a, b, e} is a subset of A.

(vii) False, ∵ {a, b, e} is a subset of A. so it does not belong to A.

(viii) False, ∵ {a, b, c} is not a subset of A.

(ix) False, ∵ ∮ is a subset and not an element of A.

(x) False, ∵ ∮ and not {∮} is a subset of A.

Sets – Exercise 1.4 – Q.7

(i) False, ∵ 1 is not an element of A.

(ii) False, ∵ {1,2,3} is not a subset of A, it is an element of A.

(iii) True, ∵ {6,7,8} is indeed an element of A.

(iv) True, ∵ {{4,5}} is indeed a subset of A.

(v) False, ∵ ∮ is a subset and not an element of A.

(vi) True, ∵ ∮ is a subset of every set, and hence a subset of A.

Sets – Exercise 1.4 – Q.8

(i) True, ∵ ∮ indeed belongs to A.

(ii) True, ∵ {∮} is an element of A.

(iii) False, ∵{1} is not an element of A.

(iv) True, ∵ {2, {∮}} is a subset of A.

(v) False, ∵ 2 is not a subset of A, it is an element of A.

(vi) True, ∵ {2,{1}} is not a subset of A.

(vii) True, ∵ {{2},{1}} is not a subset of A.

(viii) True, ∵ {∮,{∮},{1,∮}} is a subset of A.

(ix) True, ∵{{∮}} is a subset of A.
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