The negation of the Boolean expression $$\sim s \vee (\sim r \wedge s)$$ is equivalent to

We begin with the Boolean expression whose negation we must find:

$$\sim s \;\vee\; (\sim r \;\wedge\; s).$$

Our objective is to compute its logical negation, that is, to evaluate

$$\sim\bigl(\,\sim s \;\vee\; (\sim r \;\wedge\; s)\bigr).$$

First, we invoke De Morgan’s law for the negation of a disjunction, which states:

$$\sim(A \;\vee\; B) \;=\; (\sim A) \;\wedge\; (\sim B).$$

Here, we identify $$A = \sim s$$ and $$B = (\sim r \;\wedge\; s).$$ Applying the law gives

$$\sim(\,\sim s \;\vee\; (\sim r \;\wedge\; s)) \;=\; \bigl(\sim(\sim s)\bigr) \;\wedge\; \bigl(\sim(\sim r \;\wedge\; s)\bigr).$$

Now, $$\sim(\sim s)$$ is simply $$s,$$ because a double negation cancels itself. Substituting, we have

$$s \;\wedge\; \bigl(\sim(\sim r \;\wedge\; s)\bigr).$$

Next, we must simplify $$\sim(\sim r \;\wedge\; s).$$ We again use De Morgan’s law, this time for the negation of a conjunction:

$$\sim(C \;\wedge\; D) \;=\; (\sim C) \;\vee\; (\sim D).$$

Here, $$C = \sim r$$ and $$D = s.$$ Therefore,

$$\sim(\sim r \;\wedge\; s) \;=\; \bigl(\sim(\sim r)\bigr) \;\vee\; (\sim s).$$

The term $$\sim(\sim r)$$ simplifies to $$r.$$ Hence,

$$\sim(\sim r \;\wedge\; s) \;=\; r \;\vee\; (\sim s).$$

Substituting this back, the overall negation becomes

$$s \;\wedge\; \bigl(r \;\vee\; (\sim s)\bigr).$$

We now distribute $$s$$ over the disjunction $$r \;\vee\; (\sim s).$$ The distributive law of Boolean algebra tells us

$$X \;\wedge\; (Y \;\vee\; Z) \;=\; (X \;\wedge\; Y) \;\vee\; (X \;\wedge\; Z).$$

Taking $$X = s, \; Y = r, \; Z = \sim s,$$ we obtain

$$\bigl(s \;\wedge\; r\bigr) \;\vee\; \bigl(s \;\wedge\; (\sim s)\bigr).$$

The term $$s \;\wedge\; (\sim s)$$ is always false (it equals $$0$$) because a proposition cannot be simultaneously true and false. Hence that term vanishes, leaving

$$s \;\wedge\; r.$$

We have therefore shown that the negation of the original expression simplifies to $$s \wedge r,$$ which is exactly Option 2.

Hence, the correct answer is Option 2.