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--- strings_and_languages [2007/05/05 19:17]
vkuncak
+++ strings_and_languages [2019/07/02 16:16]
fabien [Simple Consequences]
@@ Line 10: / Line 10: @@
 \end{equation*}
-We use centered dot for string concatenation, as in $s_1 \cdot s_2$, and we sometimes omit it, as in $s_1 s_2$ (programming language Objective Caml uses ^ to denote string concatenation; other languages often use + as an overloaded operator, but we will not use + for concatentation).
+We use centered dot for string concatenation, as in $s_1 \cdot s_2$, and we sometimes omit it, as in $s_1 s_2$ (programming language Objective Caml uses ^ to denote string concatenation; other languages often use + as an overloaded operator).
 The concatentation is an operation $\Sigma^* \times \Sigma^* \to \Sigma^*$.  It is associative, and $\epsilon$ is left and right neutral element:
@@ Line 32: / Line 32: @@
   L^* &=& \bigcup_{n \geq 0} L^n = \{ w_1 \ldots w_n \mid w_1,\ldots,w_n \in L \}
 \end{eqnarray*}
+==== Simple Consequences ====
+Observe that
+\begin{equation*}
+   \emptyset \cdot L = \{ s_1 \cdot s_2 \mid s_1 \in \emptyset \land s_2 \in L \} = \{s_1 \cdot s_2 \mid \mbox{false} \} = \emptyset
+\end{equation*}
+Similarly, $L \cdot \emptyset = \emptyset$.
+Also directly from definition follows:
+\begin{equation*}
+  \{ w_1 \} \cdot \{ w_2 \} = \{ w_1\cdot w_2 \}
+\end{equation*}