Closure of Relations and Languages

Closure of Relations

Let S be a set and R ⊆ S x S be a relation on S. The following correspondences exist:

That is, we can form new relations Ri where i is a path of length i, because, by applying R twice we can relate x and z; by applying R three times we can relate x and w, etc.
Also, even though I haven't drawn it in, (y,w) ∈ R2 because R2 is all paths of length 2.
In general, (x0,xn) ∈ Rn iff there exists x0,x1,x2,...xn-1,xn ∈ S such that (x0,x1),(x1,x2),...,(xn-1,xn) ∈ R iff there exists a path in G:
Now, is called the transitive closure of R.

Thus:

  1. R ⊂ R+
  2. (x,y) ∈ R+ and (y,z) ∈ R -> (x,z) ∈ R+.
  3. Nothing is in R+ except by virtue of 1. and 2.
The reflexive transitive closure of R is
This differs from the transitive closure in that we add
R0 = {(a,a) | a ∈ S}

More Examples:

Closure of Languages (Sets)

Let L1 and L2 be languages (i.e., sets of strings). Then we define the concatenation, or product of L1 and L2, denoted L1L2:
L1L2 = {xy | x ∈ L1 and y ∈ L2}

Specifically, let ∑ be an alphabet (so ∑ is a language.)
Then ∑ ∑ = ∑2 = {a1a2 | a1, a2 ∈ ∑ }
and, in general,
∑n = {wa | w ∈ ∑n-1 and a ∈ ∑}

Now we define the following closures of ∑ under concatenation:

Remark: any string over ∑ of length >=0 is in ∑*, therefore a language over ∑ is a subset of ∑*.