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Question

Which of the following statements are true ? Give reason to support your answer.

(i) For any two sets A and B either A ⊆ B or B ⊆ A.

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

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

(iv) Every set has a proper subset.

(v) {a,b,a,b,a,b....} is an infinite set.

(vi) {a, b, c} and {1, 2, 3} are equivalent sets.

(vii) A set can have infinitely many subsets.

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Solution

(i) False, because the two sets A and B need to be comparable.

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

(iii) True, because 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 , because the empty set ϕ has no proper subset.

(v) False, because {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 a, has the cardinal number 2n.

If the cardinal number of A is infinite, then the cardinal of P(A) is also infinite.

Hence, the above statement is true provided the set is infinite.

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