**Question:**

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.

**Solution:**

(i) False

It is not necessary that for any two sets $A \& B$, either $A \subseteq B$ or $B \subseteq A$.

It is not satisfactory always.

Let:

$A=\{1,2\} \& B=\{\alpha, \beta, \gamma\}$

Here, neither $A \subseteq B$ nor $B \subseteq A$

(ii) False

$A=\{-1,0,1,2,3\}$ is a finite set that is a subset of infinite set $Z$.

(iii) True

$E$ very subset of a finite set is a finite set.

(iv) False

$\phi$ does not have a proper subset.

(v) False

$\{a, b, a, b, a, b, \ldots\}$ will be equal to $\{a, b\}$, which is a finite set.

(vi) True

$\{a, b, c\}$ and $\{1,2,3\}$ are equivalent sets because the number of elements in both the sets are same.

(vii) False

In the set $A=\{1,2\}$, subsets can be $\{\phi\},\{1\}$ and $\{2\}$, which are finite.