If $$(p \wedge \sim q) \wedge (p \wedge r) \rightarrow \sim p \vee q$$ is false, then the truth values of p, q and r are respectively:

We are given the compound statement $$\bigl((p \wedge \sim q) \wedge (p \wedge r)\bigr)\;\rightarrow\;(\sim p \,\vee\, q)$$ and we are told that this entire implication is false.

First, recall the fundamental truth-table rule for an implication:

$$A \rightarrow B$$ is false exactly when $$A$$ is true and $$B$$ is false.

Applying this rule, let us denote

$$A = (p \wedge \sim q) \wedge (p \wedge r), \qquad B = \sim p \,\vee\, q.$$

Because the whole implication is false, we must have

$$A \text{ is true} \quad\text{and}\quad B \text{ is false.}$$

We analyse these two conditions one after another.

Condition 1: antecedent $$A$$ is true.

Write $$A$$ more clearly:

$$A = (p \wedge \sim q) \wedge (p \wedge r).$$

For a conjunction to be true, every part must be true. So we need

$$p \wedge \sim q \text{ is true} \quad\text{and}\quad p \wedge r \text{ is true.}$$

Each of these is itself a conjunction, so we split further:

• From $$p \wedge \sim q$$ being true we get

$$p \text{ is true} \quad\text{and}\quad \sim q \text{ is true.}$$

• From $$p \wedge r$$ being true we get

$$p \text{ is true} \quad\text{and}\quad r \text{ is true.}$$

Combining all these sub-conditions we have so far

$$p = \text{T}, \quad \sim q = \text{T}, \quad r = \text{T}.$$

The statement $$\sim q = \text{T}$$ immediately gives

$$q = \text{F}.$$

Condition 2: consequent $$B$$ is false.

Now look at

$$B = \sim p \,\vee\, q.$$

For a disjunction $$X \vee Y$$ to be false, both $$X$$ and $$Y$$ must be false. Therefore

$$\sim p = \text{F} \quad\text{and}\quad q = \text{F}.$$

We already have $$q = \text{F}$$ from the first condition, so that part is consistent. The requirement $$\sim p = \text{F}$$ means

$$p = \text{T},$$

which also matches what we obtained earlier. Thus both conditions are satisfied simultaneously by exactly one set of truth values:

$$p = \text{T}, \quad q = \text{F}, \quad r = \text{T}.$$

These correspond to the option <T, F, T>.

Hence, the correct answer is Option B.