identities of regular expression

Question

how  $\phi \cdot R = R \cdot \phi = \phi$ ? where R is regular expression, and why is $\phi^* is $\epsilon$

Answer 1

Basics first. The definition of concatenation says

$R \cdot A = \{ab | a \in R \ and \ b \in A\}$

Now, if $A = \phi$, we can never form a string $ab | a \in R \ and \ b \in A$ since no matter what we do, second condition will always be false. Thus $R \cdot \phi = \phi$

Next, Kleene star definition says

$R^* = \{\epsilon\} \cup R^1 \cup R^2 \cdots$

Where $R^i$ is $i \ times$ concatenation of $R$

When $R = \phi$ for $\forall i \geqslant 1, R^i = \phi$. Thus $R^* = \epsilon$