The negation of '`sqrt5` is an integer or 5 is an irrational number' is

Options

A. `sqrt5` is not an interger and 5 is not irrational.
B. `sqrt5` is an interger and 5 irrational.
C. `sqrt5` is not an integer or 5 is not irrational.
D. `sqrt5` is irrational or 5 is an interger.

Answer

To find the negation of the statement "√5 is an integer or 5 is an irrational number," we can follow these steps:

### Step 1: Identify the components of the statement
The statement consists of two parts:
- Let P: "√5 is an integer"
- Let Q: "5 is an irrational number"

The original statement can be written as:
\[ P \lor Q \]
(where "∨" denotes "or").

### Step 2: Apply the negation
The negation of a statement of the form \( P \lor Q \) can be expressed using De Morgan's laws, which state:
\[ \neg (P \lor Q) = \neg P \land \neg Q \]
(where "∧" denotes "and").

### Step 3: Negate each component
Now, we need to negate each part:
- Negation of P: \( \neg P \): "√5 is not an integer"
- Negation of Q: \( \neg Q \): "5 is not an irrational number"

### Step 4: Combine the negated components
Now we combine the negated components using "and":
\[ \neg P \land \neg Q \]
This translates to:
"√5 is not an integer and 5 is not an irrational number."

### Final Answer
Thus, the negation of the statement "√5 is an integer or 5 is an irrational number" is:
"√5 is not an integer and 5 is not an irrational number."

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