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(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.
(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 ∮.
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.
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
∵ Crows are also Birds.
The given statement is 'False'.
A rectangle need not be a square.
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.
False, ∵ a ϵ P but a ∉ B
Note that {a} is an element of B which is different from the element 'a'.
(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.
(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} ≠ ∮.
(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.
(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.
(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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