The negation of `~svv(~r^^s) ` is equivalent to

Options

A. `s^^~r`
B. `s^^(r^^~s)`
C. `svv(rvv~s)`
D. `s^^r`

Explanation

To find the negation of the expression `~s ∨ (~r ∧ s)`, we will follow the logical rules of negation, conjunction, and disjunction step by step.

### Step-by-Step Solution:

1. **Identify the Expression**: We start with the expression `~s ∨ (~r ∧ s)`.

2. **Apply Negation**: We need to find the negation of the entire expression, which is `~(~s ∨ (~r ∧ s))`.

3. **Use De Morgan's Laws**: According to De Morgan's laws, the negation of a disjunction is the conjunction of the negations. Therefore, we can rewrite the expression as:
\[ ~(~s) ∧ ~(~r ∧ s) \]

4. **Simplify Negations**: The negation of `~s` is `s`. Now we need to simplify `~(~r ∧ s)`:
\[ ~(~r ∧ s) = ~(~r) ∨ ~s \]
By applying De Morgan's laws again, we get:
\[ r ∨ ~s \]

5. **Combine the Results**: Now we can combine the results from step 3 and step 4:
\[ s ∧ (r ∨ ~s) \]

6. **Distribute**: We can distribute `s` over the disjunction:
\[ (s ∧ r) ∨ (s ∧ ~s) \]

7. **Simplify**: The term `s ∧ ~s` is a contradiction and evaluates to the null set (or false). Therefore, we can simplify the expression to:
\[ s ∧ r \]

8. **Final Result**: The negation of `~s ∨ (~r ∧ s)` is equivalent to:
\[ s ∧ r \]

### Final Answer:
The negation of `~s ∨ (~r ∧ s)` is equivalent to `s ∧ r`.