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A multiple choice question asks this:

Number of nonequivalent propositions that only consist of $P_1, P_2, P_3$ and use the $\iff$ logical operator is?$$A)7\text{ }B)8\text{ }C)1\text{ }D)16$$

I am really uncertain about what the question is asking. Is it asking me to find all the possible propositions using the $P_1, P_2, P_3$ terms and logical operators (e.g. $\land, \lor, \neg$...) and see which ones have the same value?

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    $\begingroup$ Hint: this connective is the same as XNOR. $\endgroup$
    – Yuval Filmus
    Apr 6, 2021 at 15:04

1 Answer 1

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You just need to use $P_1$, $P_2$, $P_3$ and $\Leftrightarrow$ symbols, for example $P_2$ is a formula, $(P_1 \Leftrightarrow P_3) \Leftrightarrow P_2$ is a formula. As stated in the question, you must not use $\vee$, $\wedge$ or $\neg$.

You need to find the number of non-equivalent formulae. For example, $P_1 \Leftrightarrow P_2$ is equivalent to $P_2\Leftrightarrow P_1$, but $P_1\Leftrightarrow P_2$ is not equivalent to $P_1 \Leftrightarrow P_3$.

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  • $\begingroup$ fair enough, I am actually trying this and will update you what answer I come up with. $\endgroup$
    – Titan
    Apr 6, 2021 at 14:47
  • $\begingroup$ How is $P_1$ equivalent to $P_1\iff P_1$ though? the truth table values wouldn't match, right? $\endgroup$
    – Titan
    Apr 6, 2021 at 15:05
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    $\begingroup$ @Titan My mistake, I wrote another example. $\endgroup$
    – Nathaniel
    Apr 6, 2021 at 15:11
  • $\begingroup$ How many formulae should I be able to make? $P_1\iff P_2, P_1\iff P_3, P_2\iff P_3$ are nonequivalent, $P_1\iff P_1$ is another one which would be equivalent to $P_2\iff P_2$ and $P_3\iff P_3$, $P_1\iff P_2\iff P_3$ and maybe the ones with a single term e.g. $P_1$. Is there more? $\endgroup$
    – Titan
    Apr 6, 2021 at 15:27
  • $\begingroup$ That seems right. The other formulae seem to be equivalent to one you listed. $\endgroup$
    – Nathaniel
    Apr 6, 2021 at 15:35

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