Determine whether or not the following claims are true for a

Determine whether or not the following claims are true for all regular expressions r1 and r2, and state why. The symbol == stands for equivalence of regular expressions in the sense that both expressions denote the same language:

(a) (r₁*)* == r₁*

(b) r₁*(r₁ + r₂)* == (r₁ + r₂)*

(c) (r₁ + r₂)* == (r₁*r₂*)*

(d) (r₁r₂)* == r₁*r₂*

Solution

Post one more question to get the remaining two parts. Thanks

a) The first expression is an equivalence expression since r1* represents the continous string patter of r1

(r1...r1...r1)* = r1r1...r1....r1

Similarly r1* = r1r1....r1....r1

Hence both the classes are equal

b) The second expression is not an equivalence class since the second expression will be like

r1...r1....r1...(r1+2)....(r1+r2)......(r1+r2)

whereas the expression on the right hand side is equal to continuous patter of r1+r2 string concatentation,

r1+r2.... r1+r2.....r1+r2

Hence they both are not equivalence classes