Let $$p, q, r$$ be three logical statements. Consider the compound statements
$$S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$$ and $$S_2 : p \to (q \vee r)$$
Then, which of the following is NOT true?

We are given the formulas $$S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$$ and $$S_2 : p \to (q \vee r)$$.

To simplify $$S_1$$, we use the associative and idempotent laws of disjunction: $$S_1 = (\sim p \vee q) \vee (\sim p \vee r) = \sim p \vee q \vee \sim p \vee r = \sim p \vee q \vee r.$$

Next, the implication in $$S_2$$ is equivalent to a disjunction, namely $$S_2 = \sim p \vee (q \vee r) = \sim p \vee q \vee r.$$

Since both $$S_1$$ and $$S_2$$ simplify to the same expression $$\sim p \vee q \vee r$$, they are logically equivalent: $$S_1 \equiv S_2$$.

We now evaluate each option:

Option A: “If $$S_2$$ is True, then $$S_1$$ is True.” This holds because $$S_1 \equiv S_2$$.

Option B: “If $$S_2$$ is False, then $$S_1$$ is False.” This also holds by equivalence.

Option C: “If $$S_2$$ is False, then $$S_1$$ is True.” This would require different truth values for $$S_1$$ and $$S_2$$, contradicting their equivalence, so it is not true.

Option D: “If $$S_1$$ is False, then $$S_2$$ is False.” This holds by equivalence.

Therefore, the statement that is NOT true is Option C.