Language Operation Axioms

Union §

Absorption Of Union §

$A\cup A=A$

Commutativity Of Union §

$A\cup B=B\cup A$

Associativity Of Union §

$(A\cup B)\cup C=A\cup (B\cup C)=A\cup B\cup C$

Union With Empty Set §

$$\varnothing \cup A=A\cup \varnothing =A$$

Concatenation §

Associativity Of Concatenation §

$(A\circ B)\circ C=A\circ (B\circ C)=A\circ B\circ C$

Concatenating Empty Set §

$$\varnothing \circ A=\varnothing \circ A=\varnothing$$

Concatenating Empty Word §

$$ϵ\circ A=A\circ ϵ=A$$

Concatenating Zero Times §

$$A^0=\{ϵ\}$$

Kleene Star §

Idempotency Of Kleene Star §

$$(A^{\ast }{)}^{\ast }=A^{\ast }$$

Kleene Star With Empty Word Or Empty Set §

$${ϵ}^{\ast }={\varnothing }^{\ast }=ϵ$$

Funnily enough, this law can be derived from the concatenation of a language zero times and the definition of Kleene star:

We know that:

$$L^{\ast }=\bigcup _{i=0}^{\infty }{L}^i$$

Since $L^0=\{ϵ\}$, we have:

$${ϵ}^{\ast }=\bigcup _{i=0}^{\infty }={ϵ}^0\cup {ϵ}^1\cup \cdots \cup {ϵ}^{\infty }=\{ϵ\}\cup \{ϵ\}\cup \cdots =\{ϵ\}$$

And this also applies for the empty set:

$${\varnothing }^{\ast }=\bigcup _{i=0}^{\infty }={\varnothing }^0\cup {\varnothing }^1\cup \cdots \cup {\varnothing }^{\infty }=\{ϵ\}\cup \varnothing \cup \cdots =\{ϵ\}$$

Distributivity §

Distributivity of Concatenation over Union §

$$(A\cup B)\circ C=(A\circ C)\cup (A\circ B)$$

$$C\circ (A\cup B)=(C\circ A)\cup (C\circ B)$$

Distributivity of Kleene Star over Union §

$$(A\cup B{)}^{\ast }=(A^{\ast }{B}^{\ast }{)}^{\ast }$$