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.
(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.