Truth sets and set diagrams

Question

I am confused on how to produce truth sets from propositional logic statements For example:

$(p \land q) \implies \lnot r$

How would I turn this into a truth set, which could then be used to create a venn diagram.

I have produced a truth table which produced the result FTTTTTTT

Really unsure on how to approach this.

Answer 1

Any triple $P, Q, R$ satisfying $P\cap Q \subset R^c$ should work. Take for instance $P=\mathbb{N}$ the natural numbers, $Q=\{2n| n \in \mathbb{Z}\}$ the set of all even numbers and $R$ the set of all irrational numbers (like $\pi$ or $\sqrt{2}$). Then $P \cap Q$ is the set of all even numbers greater or equal than zero, and $R^c$ is $\mathbb{Q}$ the set of all rational numbers (assuming your universe is $\mathbb{R}$).

Answer 2

On truth sets for formulas of propositional logic, see :

Lipschutz, Schaum''s Outline Of Discrete Mathematics ( look for " truth set" in the index).

and

Rudolf Carnap , Introduction To Symbolic Logic And Its Applications ( where logical implication is explicitly defined via the set-theoretic notion of inclusion).

Notation: Let's denote by S(X) the truth set of the formula X. ( Here, X can be an atomic proposition as well as a compound sentence)

Definitions:

(1) the truth set of a formula is the set of all cases ( all n-tuples corresponding to a row of the truth table) in which the formula is true.

(2) a formula is valid ( is a tautology) if and only if it's truth set is equal to the universal set

(3) a formula X logically implies a formula Y ( or, if you prefer, Y is a logical consequence of X) if and only if S(X) is a SUBSET of S(Y).

Here you have 3 atomic propositions, so the universe ( universal set) has 8 elements ( that are ordered sets of 3 truth values).

Let's say ( as a convention) that P is the first sentence, that Q is is the second and that R is the third.

Then here , the universal set is the set that has as elements the following 3-tuples ( and you will recognize here the 8 rows of your truth-table) (T,T,T) (T,T,F) (T,F,T) (T,F,F) (F,T,T) (F,T,F) (F,F,T) (F,F,F)

Now, take each element ( each 3-tuple) of the universal set such that your formula is true in the case corresponding to the 3-tuple, and put all these 3-tuples together in a new set. This new set is the " truth set" of your formula.

Now if you want to use a Venn diagram to prove that your formula is not a tautology ( that is, is not valid) you have two options:

(1) look at the truth set of your formula and decide whether it's truth set is identical to the universal set; if there is at least one 3-tuple that does not belong to the truth set of your formula, then it is not a tautology;

(2) "create" a set S(P&Q) ( the truth set of the formula P&Q ), create also a set S(~R) and show that S(P&Q) it is NOT a subset of S(~R) .In other words, show using a diagram, that there is at least one 3-tuple that does belong to S(P&Q) but does not belong to S(~R)

Notice: S(P&Q) will be, however, a subset of your universe, and will have some 3-tuples from U as elements, namely the 3-tuples (TTT) and (TTF). The reason why it is not a subset of the truth set of (~R) is that (TTT) belongs to S(P&Q) but not to S(~R)