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Could anyone help me how do I generate output from following formula? please give answers in steps.

Language For each 0 ≤ i ≤ 5, bi is a proposition, which intuitively means that the i-th bit has value 1. Obviously, ¬bi means that the i-th bit does not have value 1, and thus it has value 0.

Axioms A possible (compact) representation of the finite set of binary strings is given by the following formula:

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zuby
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1 Answers1

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Given a $k$ from the outer $\large{\vee},$ the first inner parenthesis says you start with $k+1$ bits of $0$ followed by $5-k$ bits of $1$. The second says you start with $k+1$ bits of $1$ followed by $5-k$ bits of $0$. The middle $\vee$ collects these, and the outer $\large {\vee}$ collects all of these over the values of $k$. This does not get you a collection of all $6$ bit strings, it gets you a collection of all $6$ bit strings that have at most one transition between $0$ and $1$.

Ross Millikan
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  • Thanks @Ross Millikan for your response. but i have another problem. ~bi according to my understanding b has string pattern like b=101100 and "i" holds position of that string...means if i is 0 then b's zero position has 1 and 2nd has 0 if i is 1 am i right? – zuby Oct 20 '16 at 17:04
  • how do i illustrate first inner parenthesis into paper i think if i understand the parenthesis then i will solve the question – zuby Oct 20 '16 at 17:06
  • For your first comment, I don't think this formula can generate the string $101100$. I tried to explain that one piece has all the $1$s following all the $0$s and the other piece has it the other way around. The string $101100$ cannot be generated by either one. You are correct that the subscript $i$ refers to the position in the string. For the second, it depends on what you are trying to do in the paper. You could define it as $\vee (a \vee b)$ and define $a$ and $b$ as the two items, then refer to $a$. You can put braces under the part of the formula you want to refer to. – Ross Millikan Oct 20 '16 at 18:23
  • Thanks @Ross Millikan for you help...i=k+1 it will change value of i means 1+2+3+4+5 or change also k? – zuby Oct 21 '16 at 05:33
  • the first inner parenthesis will be iterate 5 times with ~bi? – zuby Oct 21 '16 at 05:44
  • The first inner parenthesis will iterate $6$ times as $k$ runs from $0$ to $5$. You will get $k+1$ zeros from the first $\wedge$ followed by $5-k$ ones from the second $\wedge$ – Ross Millikan Oct 21 '16 at 14:43