If `p:''4` is an odd number'' and q : ''`4^(3)` is an even number'' are two statements, then which of the following statements is equivalent to `~(prArrq)`?

Options

A. '4 is an odd number and `4^(3)` is an even number''
B. The negation of the statement ''4 is not an odd number of `4^(3)` is not an even number''
C. Both (''4 is an odd number and `4^(3)` is an even number'') and (The negation of the stateement ''4 is not an odd number of `4^(3)` is not an even number'')
D. '4 is an odd number and `4^(3)` is not an even number''

Answer

To solve the problem, we need to determine the equivalent statement for the expression \( \sim (p \Rightarrow q) \), where:

- \( p: 4 \text{ is an odd number} \)
- \( q: 4^3 \text{ is an even number} \)

### Step 1: Understand the implication \( p \Rightarrow q \)

The implication \( p \Rightarrow q \) can be expressed in logical terms as:
\[ p \Rightarrow q \equiv \sim p \lor q \]
This means "if \( p \) is true, then \( q \) is true" is equivalent to saying "either \( p \) is false or \( q \) is true."

### Step 2: Negate the implication

Now, we need to find \( \sim (p \Rightarrow q) \). Using the equivalence we just established:
\[ \sim (p \Rightarrow q) \equiv \sim (\sim p \lor q) \]

### Step 3: Apply De Morgan's Law

Using De Morgan's Law, we can simplify \( \sim (\sim p \lor q) \):
\[ \sim (\sim p \lor q) \equiv p \land \sim q \]
This means that the negation of the disjunction is equivalent to the conjunction of \( p \) and the negation of \( q \).

### Step 4: Substitute back the statements

Now we substitute back the original statements for \( p \) and \( q \):
- \( p \) is "4 is an odd number" (which is false).
- \( q \) is "4^3 is an even number" (which is true).

Thus, \( \sim q \) is "4^3 is not an even number" (which is false).

So, we have:
\[ p \land \sim q \equiv (\text{"4 is an odd number"}) \land (\text{"4^3 is not an even number"}) \]

### Conclusion

The final equivalent statement for \( \sim (p \Rightarrow q) \) is:
\[ \text{"4 is an odd number and 4^3 is not an even number."} \]