Question

Let R1 and R2 be regular expressions. From the list below, select all statements that are necessarily true.

A. L( R₁*∘ R₂*) = L ((R₁ ∘ R₂)*)
B. L ((R₁*)*) = L (R₁*)
C. L (R₁*U R₂*) = L ((R₁ U R₂)*)
D. L ((R₁*U R₂*)*) = L ((R₁ U R₂)*)

Answer 1

Statements A and D are necessarily true in the given regular expressions.

Step-by-step explanation:

From the options given, statement A and statement D are necessarily true.

Statement A: L(R₁*∘ R₂*) = L((R₁ ∘ R₂)*)

To understand this statement, let's break it down:

1. R₁* represents zero or more occurrences of R₁.
2. R₂* represents zero or more occurrences of R₂.
3. R₁ ∘ R₂ represents concatenation of R₁ and R₂.
4. (R₁*∘ R₂*) represents zero or more occurrences of R₁ followed by zero or more occurrences of R₂.
5. L(R) represents the language accepted by regular expression R.

Therefore, statement A is true because both sides of the equation represent the same language.

Statement D: L((R₁*U R₂*)*) = L((R₁ U R₂)*)

Similarly, we can break down this statement:

1. R₁*U R₂* represents either zero or more occurrences of R₁ or zero or more occurrences of R₂.
2. (R₁*U R₂*)* represents zero or more occurrences of either R₁ or R₂.
3. R₁ U R₂ represents the union of languages represented by R₁ and R₂.

Therefore, statement D is true because both sides of the equation represent the same language.