strings

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• Given a set $S$ and a natural number $n\in N$,
• $S^n$ is the set of length $n$ ‘strings’. Can be defined as product of $n$ copies of $S$ (i.e., $S\times S\times \cdots \times S$).”
  • $\{0,1{\}}^n$: set of all n-bit strings. $n=3$ → set is all possible combination of 0 and 1 in string of length 3
  • There are $2^n$ possible strings (8 when n = 3 above)
• $S^{\ast }$ is the set of finite length ‘strings’$S^{\ast }=S^0\cup S^1\cup S^2\cup ...$, where $S^0$ contains empty string.”
  • $\{0,1{\}}^{\ast }$: set of all finite-length bit strings. $\ast$ = any length, even 0. → set is all possible combinations of 0 and 1 for all possible lengths, including "", “0”, and so on
• $[n]$ is the set $\{0,1,2,\cdots ,n-1\}$