# What is the Number of Possible Nonequivalent Propositions with $P_1, P_2, P_3$ Using $\iff$ Operator?

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?

• Hint: this connective is the same as XNOR. Apr 6, 2021 at 15:04

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 $$P_1 \Leftrightarrow P_3$$.
• How is $P_1$ equivalent to $P_1\iff P_1$ though? the truth table values wouldn't match, right? Apr 6, 2021 at 15:05
• 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? Apr 6, 2021 at 15:27