The statement $$(p \wedge (\sim q)) \vee ((\sim p) \wedge q) \vee ((\sim p) \wedge (\sim q))$$ is equivalent to ____

The expression $$(p \wedge \sim q) \vee (\sim p \wedge q) \vee (\sim p \wedge \sim q)$$.

Let us simplify: Factor out from the last two terms:

$$= (p \wedge \sim q) \vee [\sim p \wedge (q \vee \sim q)]$$

$$= (p \wedge \sim q) \vee [\sim p \wedge T]$$

$$= (p \wedge \sim q) \vee \sim p$$

Using distributive law: $$A \vee (B \wedge C) = (A \vee B) \wedge (A \vee C)$$:

$$= (\sim p \vee p) \wedge (\sim p \vee \sim q)$$

$$= T \wedge (\sim p \vee \sim q)$$

$$= \sim p \vee \sim q$$

This is equivalent to $$\sim p \vee \sim q$$, which is option 2.