$\sim p$ ,$\sim\sim\sim\sim\sim p$ , and $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p$ . Which of these are equal?

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I made an attempt on this question. Please guide me if its wrong.

Consider the following boolean fuctions: $\sim p$ ,$\sim\sim\sim\sim\sim p$ , and $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p$ . Which of these are equal?

$\sim p$ and $\sim\sim\sim\sim\sim p$ are equal since if we write $\sim\sim p$, then this can be simplified to give $p$ as double negation canceled out or we can say $\sim\sim p=p$. Thus, in $\sim\sim\sim\sim\sim p$, there are $2$ pairs of double negation ($\sim$) which cancel out leaving the answer to $\sim p$. If we compare $\sim p$ and $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p$, then we can say that they are not equal since $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p$ has $6$ pairs of double negations ($\sim$) which cancel out leaving the answer as only $p$. If we compare $\sim p$ and $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p$, then we again say that they are not equal, as shown above $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p=p$. Thus, $\sim p$ is not equal to $p$.

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We have the following (this is the clearest way to see it IMO):

  • $\sim p\equiv\,\sim p$
  • $\sim\sim\sim\sim\sim p\equiv\,\sim(\sim\sim(\sim\sim p))\equiv\,\sim p$
  • $\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim\sim p\equiv (\sim\sim(\sim\sim(\sim\sim(\sim\sim(\sim\sim (\sim\sim p))))))\equiv p$

Hence, the first two are equal. Your answer is correct, as MJD has pointed out in a comment. The above is largely provided to give increased clarity.