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Venn Diagram Image

I am struggling to understand truth sets and the symbols used.

In this context $P, Q, R$ are truth sets of $p,q,r$.

I am unsure of how to find the truth set of this expression $$ (p ∧ q) \to \neg r $$ And how to represent in a venn diagram

Would the diagram be something like this?

Andrés E. Caicedo
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Lyhbm
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    Hint: Convert the implication to a disjunction via the identity $p\to q\iff \neg p\lor q$. – John Douma Feb 20 '19 at 19:19
  • Am I along the right lines with the venn diagram? – Lyhbm Feb 20 '19 at 19:25
  • I don't know. I didn't compute it. I just know that if you do the conversion you will be working with "ands" and "ors" which are easy to express with Venn diagrams. – John Douma Feb 20 '19 at 19:27
  • How do I express in terms of truth sets? So (p ∪ (q ∩ r)) etc? – Lyhbm Feb 20 '19 at 19:31
  • If, by truth set, you mean the set of values that satisfy the expression then you can construct the truth table and see which values of $p$, $q$ and $r$ lead to true outcomes for the expression. In fact, you can do that without converting the implication and construct the Venn diagram from that. – John Douma Feb 20 '19 at 19:34
  • I am fine with truth tables and the venn diagram, just unsure of truth sets. So for the above expression, I am unsure whether it is P∩Q∩R or something else – Lyhbm Feb 20 '19 at 19:38
  • Your diagram cannot be correct because you have drawn $p\land q\land r$ but your expression says that an element in $p\land q$ cannot be in $r$. – John Douma Feb 20 '19 at 19:39
  • So what would you suggest it would be? – Lyhbm Feb 20 '19 at 19:46
  • Try one or both of the suggestions above. You'll get it. – John Douma Feb 20 '19 at 20:55
  • I have tried both but am still unclear. – Lyhbm Feb 20 '19 at 21:42

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See Lipschutz, Schaum's Outline Of Discrete mathematics ( Index : " truth set").

The truth set of a proposition is the set of all cases in which this proposition is true.

Here, the " cases" are represented by lists ( ordered sets) of truth-values ( True/False): couples of truth-values if your formula has 2 atomic propositions, triples of truth-values if it has 3, etc.

So the truth set of a formula is a set of couples, or a set of triples, etc. ( depending on the number of atomic propositions involved in the formula).

To construct your Venn diagram, use as U ( universal set) the set of all possible couples, triples, etc.

Example ( with, by convention, P as first sentence and Q as second sentence)

The truth set of (P&Q) is { (T,T)} ( a set with only one element, for (P&Q) is true in only one case, namely the case in which P is true and Q is also true.

The truth set of (P v Q) is { (T,T), (T,F), (F,T)}

etc.

The formula (P&Q) => Q is valid ( is a tautology) since it's truth set is : 

                 { (T,T), (T,F), (F,T), (F,F) }

in other words, the formula is valid, since it's truth set is U, the universal set itself ( for a formula involving only 2 atomic sentences)!