Length of Concatenation
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Theorem
Let $S$ and $T$ be words, and let $ST$ be their concatenation.
Then:
- $\size {S T}\ = \size S + \size T$
where $\size S$ denotes the length of $S$.
Proof
Because of the unique readability of $ST$, we can determine for each symbol $s$ that is part of $S T$, whether:
- $s$ is part of $S$
- $s$ is part of $T$
and furthermore, precisely one of these options occurs.
There are $\size S$ symbols in $S$, and $\size T$ symbols in $T$.
In total, then, $S T$ is seen to consist of $\size S + \size T$ symbols.
$\blacksquare$