Negation of "3 is an odd number and 7 is a rational number is-

Options

A. 3 is not and odd number and 7 is not a rational number
B. 3 is odd number of 7 is a rational number
C. 3 is an odd number or 7 is not a rational number
D. 3 is not an odd number or 7 is not a rational number.

Answer

To find the negation of the statement "3 is an odd number and 7 is a rational number," we can follow these steps:

### Step 1: Identify the components of the statement
The given statement can be broken down into two parts:
- Let \( p \) be the statement "3 is an odd number."
- Let \( q \) be the statement "7 is a rational number."

### Step 2: Write the statement in logical form
The statement can be expressed in logical form as:
\[ p \land q \]
This means "p and q," which is the conjunction of both statements.

### Step 3: Apply the negation
To negate the conjunction \( p \land q \), we use De Morgan's Laws, which state:
\[ \neg (p \land q) = \neg p \lor \neg q \]
This means that the negation of "p and q" is equivalent to "not p or not q."

### Step 4: Negate each component
Now we need to negate each component:
- The negation of \( p \) ("3 is an odd number") is:
\[ \neg p: \text{"3 is not an odd number."} \]
- The negation of \( q \) ("7 is a rational number") is:
\[ \neg q: \text{"7 is not a rational number."} \]

### Step 5: Combine the negated components
Now we combine the negated components using "or":
\[ \neg p \lor \neg q \]
This translates to:
"3 is not an odd number or 7 is not a rational number."

### Final Answer
Thus, the negation of the statement "3 is an odd number and 7 is a rational number" is:
**"3 is not an odd number or 7 is not a rational number."**

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