# Logic and Sets Expressions

The logical operations $\Rightarrow$ and $\Leftrightarrow$ correspond to relations on sets. What are these two relations? Is there anything in logic that could not be expressed in terms of set theory and vice versa?

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To say that $P(x)\Rightarrow Q(x)$ is to say that if $x$ satisfies property $P$ then it satisfies property $Q$.

One can see this as $x\in P\Rightarrow x\in Q$, or in a compact notation: $P\subseteq Q$.

From this you can infer that $P\Leftrightarrow Q$ corresponds to $P\subseteq Q\land Q\subseteq P$, which essentially to say $P=Q$.

Similarly, if we think of $P(x)$ as $x\in P$, we have that $P(x)\land Q(x)$ is the same as to say $x\in P$ and $x\in Q$, or $x\in P\cap Q$; the $\lor$ corresponds the same way with $P\cup Q$; and lastly $\lnot P(x)$ is to say that $x\notin P$.

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"$P\cap Q$" and "$P\cup Q$" are sets, but "$x\not\in P$", "$P\subseteq Q$", and "$P=Q$" are logical formulas, so in the last three cases you haven't actually expressed the original formula as a set, just as a different formula. – Henning Makholm Dec 16 '11 at 13:28

I will make a short story.... very long. I hope you will find it interesting/illuminating, but if you cannot really bear it, jump right to the last few lines for a summary.