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--- mathematics:languagesandgrammar [2024/10/09 14:40] – created tami
+++ mathematics:languagesandgrammar [2024/10/09 15:12] (current) – [Closure of an alphabet] tami
@@ Line 8: / Line 8: @@
 $$
   \begin{aligned}
-    L^0 &= \{\Lambda\}\newline
+    L^0 &= \{\Lambda\} \\
     L^n &= L \cdot L^{n-1}, \text{if} \ n > 0
   \end{aligned}
 $$
+**Example.** If $L = \{a,bb\}$ then the first few powers of $L$ are...
+$$
+  \begin{aligned}
+    L^0 &= \{\Lambda\} \\
+    L^1 &= L = \{a,bb\} \\
+    L^2 &= L \cdot L = \{aa,abb,bba,bbbb\} \\
+    L^3 &= L \cdot L^2 = \{aaa,aabb,abba,abbbb,bbaa,bbabb,bbbba,bbbbbb\}
+  \end{aligned}
+$$
+==== Closures of a Language ====
+If L is a language over $\Sigma$ (i.e. $L \subset \Sigma^{\ast}$) then **the closure** of $L$ is denoted $L^{\ast}$, and the **positive closure** of $L$ is denoted $L^+$.
+$$
+  \begin{aligned}
+    L^{\ast} &= L^0 \cup L^1 \cup L^2 \cup ... \\
+    L^+ &= L^1 \cup L^2 \cup L^3 \cup ...
+  \end{aligned}
+$$
+==== Closure of an alphabet ====
+The closure of $\Sigma$ coincides with out definition of $\Sigma^{\ast}$ as the set f all string over $\Sigma$. In other words, we have a nice representation of $\Sigma^{\ast}$ as follows:
+$$
+  \Sigma^{\ast} = \Sigma^0 \cup \Sigma^1 \cup \Sigma^1 \cup ...
+$$
+Where $\Sigma^k$ denotes the set of strings of length $k$, each of whose symmbols in $\Sigma$.
+===== Grammar =====