If $$p, q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p$$, $$q$$ and $$r$$ makes the logical expression $$\{(p \vee q) \wedge ((\neg p) \vee r)\} \to ((\neg q) \vee r)$$ false?

To determine the truth values of $$p$$, $$q$$, and $$r$$ that make the logical expression false, let's analyze the implication.

The given expression is:

$$\{(p \lor q) \land (\neg p \lor r)\} \to (\neg q \lor r)$$

Let the premise be $$A = (p \lor q) \land (\neg p \lor r)$$and the conclusion be $$B = \neg q \lor r$$.

The expression is of the form $$A \to B$$.

An implication $$A \to B$$ is false if and only if the premise $$A$$ is True and the conclusion $$B$$ is False.

Step 1: Make the ($$B$$) False

We need $$B = \neg q \lor r$$ to be False.

For a logical OR ($$\lor$$) to be false, both of its components must be false.

$$\neg q$$ must be False $$ \implies $$q is True

$$r$$ must be False $$ \implies$$ r is False

So, we have established that $$q = True $$ and $$r = False$$.

Step 2: Ensure the premise ($$A$$) is True

We need $$A = (p \lor q) \land (\neg p \lor r)$$ to be True.

Substitute the known values ($$q = \text{True}$$, $$r = \text{False}$$) into the premise:

$$p \lor q = p \lor \text{True} = \text{True}$$(since $$q$$ is True, the OR statement is True regardless of $$p$$)

$$\neg p \lor r = \neg p \lor \text{False} = \neg p$$

For the premise $$A$$ to be True, both parts of the AND ($$\land$$) statement must be True.

$$\text{True} \land \neg p$$ must be True.

This means $$\neg p$$ must be True, which implies p is False .

Conclusion

The combination of truth values that makes the given logical expression false is:

$$p = \text{False}$$

$$q = \text{True}$$

$$r = \text{False}$$