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.

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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