Question

Suppose that \({\bf{A}}\) and \({\bf{B}}\) are subsets of \({\bf{V}}\), where \({\bf{V}}\) is an alphabet. Does it follow that \({\bf{A}} \subseteq {\bf{B}}\) if \({\bf{A}} \subseteq {{\bf{B}}^{\bf{*}}}\) ?

Short answer

No, the given does not follow \({\bf{A}} \subseteq {\bf{B}}\) if \({\bf{A}} \subseteq {{\bf{B}}^{\bf{*}}}\).

Step-by-step solution

Definition

In mathematics, set\({\bf{A}}\)is a subset of a set\({\bf{B}}\)if all elements of\({\bf{A}}\)are also elements of\({\bf{B}}\);\({\bf{B}}\)is then a superset of\({\bf{A}}\). It is possible for\({\bf{A}}\)and\({\bf{B}}\)to be equal; if they are unequal, then\({\bf{A}}\)is a proper subset of\({\bf{B}}\).

Check if the given statement is true or not

This is clearly not necessarily true. For example, it can take \({\bf{A = }}{{\bf{V}}^{\bf{*}}}\) and \({\bf{B = V}}\). Then \({{\bf{A}}^{\bf{*}}}\) is again \({{\bf{V}}^{\bf{*}}}\), so it is true that \({{\bf{A}}^{\bf{*}}} \subseteq {{\bf{B}}^{\bf{*}}}\), but of course \({\bf{A}} \not\subseteq {\bf{B}}\) (for one thing, \({\bf{A}}\) is infinite and \({\bf{B}}\) is finite).

Therefore, the given does not follow \({\bf{A}} \subseteq {\bf{B}}\) if \({\bf{A}} \subseteq {{\bf{B}}^{\bf{*}}}\).