Is the set of all possible binary strings countable?

Question

Determine whether the given sets are countable or uncountable. Justify your answers either with bijections or using results/methods.

The set of all possible binary strings. Note a binary string is a sequence of 0s and 1s.

If a binary string consists only of 0s and 1s, then is it even possible to find a bijection for it?

Finite binary strings or infinite binary strings or both? – kimchi lover Dec 11 '17 at 02:26 — kimchi lover, Dec 11 '17 at 02:26
@kimchilover Doesn't matter, I've answered both ways. – Parcly Taxel Dec 11 '17 at 02:34 — Parcly Taxel, Dec 11 '17 at 02:34

score 6 · Answer 1 · answered Dec 11 '17 at 02:32

6

If the strings are of finite length, they are countable. A bijection from them to the set of positive natural numbers is given by $f(s)=1s_2$, where $1s$ denotes prepending 1 to $s$.

If the strings are of infinite length, they are uncountable. The bijection of prepending a radix point turns them into the set of real numbers in $[0,1]$, which were themselves shown uncountable by Cantor.

answered Dec 11 '17 at 02:32

Parcly Taxel

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I'm not sure prepending a radix point to an infinite binary string gives you a bijection to $[0,1]$ but that doesn't matter, seeing as Cantor's theorem is simpler to prove for binary strings than for real numbers. – bof Dec 11 '17 at 04:04

Is the set of all possible binary strings countable?

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