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Let us define a set S of biry strings according to the following rules:

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Let us define a set S of binary strings according to the following rules:

Base: The empty string and the string 1 are in S.

Recursion: If xεS, then so are x0 and x11.  (That is, x followed by 0 or 11.)

If S contains no other strings, which of the following strings are in S?

111,10101,11,1101,1011,010 or 00000

Apr 18th, 2015

a) 111 ϵ S, because 1 ϵ S and x ϵ S  implies x11 ϵ S

b) 10101 not ϵ S, because if the string ends with 1 and its length is more than 1, then it must end with 11.

c) 11 ϵ S because the empty string is in S and x ϵ S → x11 ϵ S

d) 1101 not ϵ S, see the example b). 

e) 1011 ϵ S, because 1 ϵ S,  10 ϵ S, and 1011 ϵ S.

f) 010 not ϵ S. Assume the contrary. Then 01 ϵ S but (see the example b)). So, we get a contradiction q.e.d.

g) 00000 ϵ S because empty string is in S, therefore, 0 ϵ S, 00 ϵ S, 000 ϵ S etc.

Apr 18th, 2015

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Apr 18th, 2015
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