Question

Show that if A is a set of strings, then\({\bf{A}}\emptyset {\bf{ = }}\emptyset {\bf{A = }}\emptyset \).

Short answer

The following steps show that\({\bf{A}}\emptyset {\bf{ = }}\emptyset {\bf{A = }}\emptyset \).

Step-by-step solution

Show the result.

Here given that A is a set of string.

\({\bf{A}}\emptyset \)represents the concatenation of A and\(\emptyset \). Which contains all strings of the form xy where x is string of A sand y is the string of∅.

\({\bf{A}}\emptyset {\bf{ = \{ xy|x}} \in {\bf{A}}\,{\bf{and}}\,{\bf{y}} \in \emptyset \} \)

Since the statement \({\bf{y}} \in \emptyset \) is not true, the statement \({\bf{x}} \in {\bf{A}}\)is also not true. Thus\({\bf{A}}\emptyset {\bf{ = }}\emptyset \).

show the other result.

\(\emptyset \) A represents the concatenation of\(\emptyset \)and A, which contains all strings of the form xy where x is string of \(\emptyset \)and y is the string of A.

\(\emptyset {\bf{A = \{ xy|x}} \in \emptyset \,{\bf{and}}\,{\bf{y}} \in {\bf{A\} }}\)

Since the statement \({\bf{x}} \in \emptyset \) is not true, the statement \({\bf{y}} \in {\bf{A}}\) is also not true. Thus \(\emptyset {\bf{A = }}\emptyset \).

Thus \({\bf{A}}\emptyset {\bf{ = }}\emptyset {\bf{A = }}\emptyset \).

Therefore, this shows that \({\bf{A}}\emptyset {\bf{ = }}\emptyset {\bf{A = }}\emptyset \).