Understanding the Problem

We are asked to find the negation of the logical expression: ~ s ∨ (~ r ∧ s). We need to simplify this negation and find which of the given options it is equivalent to.

Step 1: Write Down the Original Expression and Its Negation

Original Expression: $\neg s\vee (\neg r\wedge s)$

Negation of the Expression: $\neg [\neg s\vee (\neg r\wedge s\left)\right]$

Step 2: Apply De Morgan's Law

De Morgan's Law states that the negation of a disjunction is the conjunction of the negations:

$\neg (A\vee B)\equiv \neg A\wedge \neg B$

Applying this to our expression, where A is ¬s and B is (¬r ∧ s):

$\neg [\neg s\vee (\neg r\wedge s\left)\right]\equiv \neg (\neg s)\wedge \neg (\neg r\wedge s)$

Step 3: Simplify the Negations

First, simplify ¬(¬s):

$\neg (\neg s)\equiv s$ (The negation of a negation is the original proposition).

Now, simplify ¬(¬r ∧ s). We apply De Morgan's Law again, this time for a conjunction:

$\neg (A\wedge B)\equiv \neg A\vee \neg B$

Applying this, where A is ¬r and B is s:

$\neg (\neg r\wedge s)\equiv \neg (\neg r)\vee \neg s\equiv r\vee \neg s$

Step 4: Combine the Results

Now we substitute our simplifications back into the expression from Step 2:

$s\wedge (r\vee \neg s)$

Step 5: Apply the Distributive Law

We can now distribute s over the disjunction (r ∨ ¬s):

$s\wedge (r\vee \neg s)\equiv (s\wedge r)\vee (s\wedge \neg s)$

Step 6: Simplify Using the Complement Law

The term (s ∧ ¬s) is a contradiction. According to the Complement Law, a proposition AND its negation is always false:

$s\wedge \neg s\equiv False \left(0\right)$

Furthermore, anything OR False is just the original thing (Identity Law):

$(s\wedge r)\vee False\equiv s\wedge r$

Final Answer

The negation of ~ s ∨ (~ r ∧ s) simplifies to $s\wedge r$.

Therefore, the correct equivalent expression from the options is: $s\wedge r$

Related Topics & Formulae

Key Logical Equivalences Used:

• Double Negation: $\neg (\neg p)\equiv p$
• De Morgan's Laws:
  • $\neg (p\wedge q)\equiv \neg p\vee \neg q$
  • $\neg (p\vee q)\equiv \neg p\wedge \neg q$
• Distributive Law: $p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)$
• Complement Law: $p\wedge \neg p\equiv False \left(0\right)$
• Identity Law: $p\vee False\equiv p$

Related Concepts:

Propositional Logic: The branch of logic that deals with propositions (statements that are either true or false) and their combinations using logical operators like AND (∧), OR (∨), and NOT (¬).

Logical Equivalence: Two compound propositions are logically equivalent if they have the same truth value for all possible combinations of truth values of their variables. The symbol ≡ is used to denote equivalence.

Simplification: The process of using logical equivalences to rewrite a complex proposition into a simpler, equivalent form. This is crucial for designing efficient digital circuits and constructing mathematical proofs.