Logical equivalent propostion to the proposition ` ~ ( p ^^ q)` is

Options

A. ` ~ p ^^ ~ q`
B. ` ~ p vv ~ q`
C. ` ~ p to ~ q`
D. ` ~ p harr ~ q`

Explanation

To find the logical equivalent proposition to the proposition \( \sim (p \land q) \), we can use De Morgan's laws. Here’s a step-by-step solution:

### Step 1: Understand the given proposition
We have the proposition \( \sim (p \land q) \), which is the negation of the conjunction of \( p \) and \( q \).

### Step 2: Apply De Morgan's Law
According to De Morgan's laws, the negation of a conjunction can be expressed as the disjunction of the negations. Specifically, De Morgan's law states:
\[ \sim (A \land B) \equiv \sim A \lor \sim B \]
In our case, \( A \) is \( p \) and \( B \) is \( q \).

### Step 3: Substitute into De Morgan's Law
Applying De Morgan's law to our proposition:
\[ \sim (p \land q) \equiv \sim p \lor \sim q \]

### Step 4: Conclusion
Thus, the logical equivalent proposition to \( \sim (p \land q) \) is:
\[ \sim p \lor \sim q \]

### Final Answer
The logical equivalent proposition is \( \sim p \lor \sim q \).

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