Question

Which of the following statements are true? Give reason to support your answer.
(i) For any two sets A and B either $A\subseteq B\mathrm{or}B\subseteq A;$
(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.

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{False} \\ \mathrm{It}\mathrm{is}\mathrm{not}\mathrm{necessary}\mathrm{that}\mathrm{for}\mathrm{any}\mathrm{two}\mathrm{sets}A\&B,\mathrm{either}A\subseteq BorB\subseteq A. \\ \mathrm{It}\mathrm{is}\mathrm{not}\mathrm{satisfactory}\mathrm{always}. \\ Let: \\ A=\{1,2\}\&B=\{\alpha ,\beta ,\gamma \} \\ \mathrm{Here},\mathrm{neither}A\subseteq B\mathrm{nor}B\subseteq A. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{False} \\ A=\{-1,0,1,2,3\}\mathrm{is}\mathrm{a}\mathrm{finite}\mathrm{set}that\mathrm{is}\mathrm{a}\mathrm{subset}\mathrm{of}\mathrm{infinite}\mathrm{set}Z. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{True} \\ E\mathrm{very}\mathrm{subset}\mathrm{of}\mathrm{a}\mathrm{finite}\mathrm{set}\mathrm{is}\mathrm{a}\mathrm{finite}\mathrm{set}. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\mathrm{False} \\ \mathrm{\varphi }\mathrm{does}\mathrm{not}\mathrm{have}\mathrm{a}\mathrm{proper}\mathrm{subset}. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)\mathrm{False} \\ \{a,b,a,b,a,b,...\}\mathrm{will}\mathrm{be}\mathrm{equal}\mathrm{to}\{a,b\},\mathrm{which}\mathrm{is}\mathrm{a}\mathrm{finite}\mathrm{set}. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vi}\right)\mathrm{True} \\ \{a,b,c\}\mathrm{and}\{1,2,3\}\mathrm{are}\mathrm{equivalent}\mathrm{sets}\mathrm{because}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{elements}\mathrm{in}\mathrm{both}\mathrm{the}\mathrm{sets}\mathrm{are}\mathrm{same}. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vii}\right)\mathrm{False} \\ \mathrm{In}\mathrm{the}\mathrm{set}A=\{1,2\},\mathrm{subsets}\mathrm{can}\mathrm{be}\left\{\varphi \right\},\left\{1\right\}\mathrm{and}\left\{2\right\},\mathrm{which}\mathrm{are}\mathrm{finite}. \end{gathered}$$