Identities of Regular Expression

Summary

This video explains various identities of regular expressions, which are crucial for solving problems and for exams like GATE. It covers basic operations like union, concatenation, and closure with empty set (Phi) and epsilon.

Highlights

Identity 5: R + R = R0:02:16

The union of a regular expression with itself results in the same regular expression.

Identity 8: (R) = R*0:03:23

Taking the closure of a regular expression's closure results in the original closure of the regular expression.

Identity 1: Phi + R = R0:00:20

The union of an empty set (Phi) and any regular expression (R) is the regular expression itself. Phi represents an empty set, and '+' denotes union.

Identity 2: (Phi . R) + (R . Phi) = Phi0:00:51

Concatenating an empty set (Phi) with a regular expression (R), or R with Phi, and then taking their union, results in an empty set (Phi).

Identity 3: Epsilon . R = R . Epsilon = R0:01:16

Concatenating epsilon (empty string) with any regular expression (R), or R with epsilon, results in the regular expression R itself.

Identity 6: R* . R* = R*0:02:28

Concatenating the closure of a regular expression with itself results in the closure of that regular expression.

Identity 9: Epsilon + R . R* = Epsilon + R* . R = R*0:03:41

This identity explains that the union of epsilon with R concatenated with R* (which is R+) is equal to R*. This effectively means adding the empty string to all possible non-empty strings formed by R gives all possible strings formed by R, including the empty string.

Identity 10: (P . Q) . P = P . (Q . P)*0:04:56

Discusses a more complex identity involving concatenation and closure of two regular expressions P and Q. This identity needs to be remembered as it is.

Identity 11: (P + Q)* = (P* . Q) = (P* + Q)0:05:15

Expands on ways to express the closure of the union of P and Q. It can be represented in multiple forms involving concatenation and closure of individual regular expressions.

Identity 12: (P + Q) . R = P . R + Q . R and R . (P + Q) = R . P + R . Q0:05:36

This identity demonstrates the distributive property of concatenation over union, similar to multiplication over addition. R concatenated with (P union Q) is equivalent to (R concatenated with P) union (R concatenated with Q).

Identity 4: Epsilon Closure = Epsilon and Phi Closure = Epsilon0:01:48

The closure of epsilon is epsilon. Importantly, the closure of an empty set (Phi) is epsilon, not Phi.

Identity 7: R . R* = R* . R0:03:01

The concatenation of a regular expression R with its closure R* is equivalent to the concatenation of R* with R.