Question

a) Define the set of regular expressions over a set I.

b) Explain how regular expressions are used to represent regular sets.

Short answer

a) The definition of regular expressionsare “The definition of a regular expression over a set \(I\) is as follows: The symbol \(\emptyset \) is a regular expression; The symbol \(\lambda \)is a regular expression; The symbol \(x\) is a regular expression whenever \(x \in I\); The symbols \(\left( {AB} \right),\left( {A \cup B} \right),\) and \(A*\) are regular expressions whenever \(A\) and \(B\) are regular expressions”.

b) Every regular set is defined by a regular expression and thus use regular expressions to indicate which strings are in the regular set.

Step-by-step solution

General form

Regular expressions (Definition):The definition of a regular expression over a set \(I\) is as follows:

A regular expression is the symbol \(\emptyset \);

A regular expression is the symbol \({\bf{\lambda }}\);

whenever \({\bf{x}} \in {\bf{I}}\), a regular expression x;

Both \(A\) and \(B\) are regular expressions, as are the regular expressions\(\left( {{\bf{AB}}} \right){\bf{,}}\left( {{\bf{A}} \cup {\bf{B}}} \right){\bf{,}}\), and \({\bf{A*}}\).

Regular set: The sets represented by regular expressions are called regular sets.

Step 2: Describe the regular expressions

Regular expressions:

The definition of a regular expression over a set I is as:

A regular expression is the symbol \(\emptyset \);

A regular expression is the symbol \(\lambda \);

whenever \({\bf{x}} \in {\bf{I}}\), a regular expression of type x;

Both A and B are regular expressions, as are the regular expressions \(\left( {{\bf{AB}}} \right){\bf{,}}\left( {{\bf{A}} \cup {\bf{B}}} \right){\bf{,}}\), and \({\bf{A*}}\).

Hence, the definition of regular expressions is shown above.

Explanation of the regular set

Regular set:

Every regular set is defined by a regular expression and thus we use regular expressions to indicate which strings are in the regular set.

Hence, the explanation of the regular set is shown above.