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Finite and Infinite Set

Every subset ...

Question

Every subset of an infinite set is infinite.?

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Solution

A set is defined as an infinite set if and only if there exists a subset whose cardinality is greater than any number of the set.
Ex; set of natural numbers etc.

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Q. Which of the following statements are true? Give reason to support your answer.
(i) For any two sets A and B either

A \subseteq B or B \subseteq 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.

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

(i) For any two sets A and B either A $\subseteq$ B or B $\subseteq$ 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.

A = {1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, . . .} .

Prove that for every positive integer

n \geq 3

the set A contain an infinite non-constant arithmetic sequence.

Q. A set of all even numbers is an infinite set.

Which of the following is an infinite set?

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