A Regular Expression can be recursively defined as follows −

ε is a Regular Expression indicates the language containing an empty string. (L (ε) = {ε})

φ is a Regular Expression denoting an empty language. (L (φ) = { })

x is a Regular Expression where L = {x}

If X is a Regular Expression denoting the language L(X) and Y is a Regular Expression denoting the language L(Y), then

X + Y is a Regular Expression corresponding to the language L(X) ∪ L(Y) where L(X+Y) = L(X) ∪ L(Y).

X . Y is a Regular Expression corresponding to the language L(X) . L(Y) where L(X.Y) = L(X) . L(Y)

R* is a Regular Expression corresponding to the language L(R*)where L(R*) = (L(R))*

If we apply any of the rules several times from 1 to 5, they are Regular Expressions.

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