Operations That Can Be Performed on Languages

Concatenation
L₁ . L₂ = L₁L₂ = {w ε Σ| w=xy for some x ε L₁ and y ε L₂}

Closure (pairing of elements)
examples:
L₁ = {a}
L₁* = {e,a,aa,aaa,aaaa,...}
L₂ = {aa, bb}
L₂* = {e,aa, bb,aaaa,aabb,bbaa,bbbb,...}
L₃ = 0
L₃* = {e}
(note: * is Kleene's Star and e represents empty)

Positive Closure Property
L⁺ = LL*
example:
L = {aa,bb}
L* = {e,aa,bb,aaaa,aabb,bbaa,bbbb,...}
L⁺ = {aa,bb,aaaa,aabb,bbaa,bbbb,...}
Elements of L are concatenated with the elements of L*.

Union
L = L₁ ∪ L₂
example:
L₁ = {a}
L₂ = {b}
L₁ ∪ L₂ = {a,b}

Intersection
L = L₁ ∩ L₂
example:
L₁ = {a,b}
L₂ = {b}
L₁ ∩ L₂ = {b}

Difference
L = L₁ - L₂
example:
L₁ = {a,b}
L₂ = {b}
L₁ - L₂ = {a}

Complement
~L = Σ* - L