How would the venn diagram for this logical expression look like?

Question

Given the expression:

(p $\implies$ q) $\wedge$ (q $\implies$ r)

I got rid of the implication to get:

(¬p $\vee$ q) $\wedge$ (¬q $\vee$ r)

I first drew 2 venn diagrams, the left hand side of the conjunction symbol then the right.

There are three sets, when I draw the 2 separately, do I draw just two circles, and then combine the two? Or do I start by drawing 3 circles, even though one set is not in the other?

If I draw the venn diagrams for both sides just using two circles, and then combine the two, I get a venn diagram with where everything outside the 3 sets are shaded. But if I started off with 3 circles, and then combine them due to the conjunction symbol, I get everything outside the 3 sets shaded and the part where all three sets share a common area.

Which one is correct?

Thanks!

Answer 1

Suggestion for simplification:

An implication $p \rightarrow q$ is analogous to the set containment $P \subseteq Q$. It tells us nothing about any arbitrary element $x$, except that if $x \in P$, then $x\in Q$. Similarly, $q \rightarrow r$ is analogous to the set containment $Q \subseteq R$. So if $x \in Q$, then $x\in R$.

So you have three concentric circles, $p$ inside $q$ inside $r$ (nested), depicting $P \subseteq Q \subseteq R$. They may very well be the same circle, but none of p can be outside q, and none of q can be outside r. Nothing need be shaded. We are only given enough information to draw how the circles (sets) of the Venn Diagram are related.

That's it.