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 find the equivalent statement for \( \sim (p \Rightarrow q) \), where:

- \( p \): "4 is an odd number"
- \( q \): "4^3 is an even number"

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

The implication \( p \Rightarrow q \) can be expressed using logical equivalences. The statement \( p \Rightarrow q \) is equivalent to \( \sim p \lor q \).

### Step 2: Negate the implication

Now, we need to negate the implication:
\[ \sim (p \Rightarrow q) = \sim (\sim p \lor q) \]

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

Using De Morgan's Law, we can rewrite the negation:
\[ \sim (\sim p \lor q) = p \land \sim q \]

### Step 4: Substitute the statements

Now we substitute back the definitions of \( p \) and \( q \):
- \( p \) is "4 is an odd number"
- \( q \) is "4^3 is an even number"

Thus, we have:
\[ p \land \sim q = \text{"4 is an odd number"} \land \sim \text{"4^3 is an even number"} \]

### Step 5: Rewrite the statement

The negation of \( q \) means "4^3 is not an even number". Therefore, we can rewrite the statement as:
\[ \text{"4 is an odd number"} \land \text{"4^3 is not an even number"} \]

### Final Statement

Thus, the equivalent statement to \( \sim (p \Rightarrow q) \) is:
\[ \text{"4 is an odd number and 4^3 is not an even number."} \]