Which of the following statements is a tautology?

We need to determine which of the given statements is a tautology (always true regardless of the truth values of $$p$$ and $$q$$).

Option A: $$p \to (p \wedge (p \to q))$$

When $$p = T, q = F$$: $$p \to q = F$$, so $$p \wedge F = F$$, and $$T \to F = F$$. Not a tautology.

Option B: $$(p \wedge q) \to (\neg p \to q)$$

Let us check all cases:

$$p = T, q = T$$: $$(T \wedge T) \to (F \to T) = T \to T = T$$ ✓

$$p = T, q = F$$: $$(T \wedge F) \to (F \to F) = F \to T = T$$ ✓

$$p = F, q = T$$: $$(F \wedge T) \to (T \to T) = F \to T = T$$ ✓

$$p = F, q = F$$: $$(F \wedge F) \to (T \to F) = F \to F = T$$ ✓

All cases give TRUE. This is a tautology!

Option C: $$(p \wedge (p \to q)) \to \neg q$$

When $$p = T, q = T$$: $$(T \wedge T) \to F = T \to F = F$$. Not a tautology.

Option D: $$p \vee (p \wedge q)$$

When $$p = F, q = F$$: $$F \vee F = F$$. Not a tautology.

The correct answer is Option B: $$(p \wedge q) \to (\neg p \to q)$$.