Negation of ' `sqrt(5)` is irrational or 3 is rational' is

Options

A. `sqrt(5)` is ratio.nal or 3 is irrational
B. `sqrt(5)` is rational and 3 is rational
C. `sqrt(5)` is rational and 3 is irrational
D. `sqrt(5)` is irrational and 3 is irrational

Answer

To find the negation of the statement "√5 is irrational or 3 is rational," we can follow these steps:

### Step 1: Identify the Statements
Let:
- P: "√5 is irrational"
- Q: "3 is rational"

### Step 2: Write the Original Statement
The original statement can be expressed as:
- P or Q (P ∨ Q)

### Step 3: Apply the Negation
The negation of a disjunction (P ∨ Q) is given by De Morgan's laws:
- ¬(P ∨ Q) = ¬P ∧ ¬Q

This means we need to negate both P and Q and change the "or" to "and."

### Step 4: Negate Each Statement
Now we find the negations of P and Q:
- ¬P: "√5 is not irrational," which means "√5 is rational."
- ¬Q: "3 is not rational," which means "3 is irrational."

### Step 5: Combine the Negated Statements
Combining these negations, we get:
- ¬(P ∨ Q) = ¬P ∧ ¬Q
- This translates to: "√5 is rational and 3 is irrational."

### Final Answer
Thus, the negation of the statement "√5 is irrational or 3 is rational" is:
- "√5 is rational and 3 is irrational."

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