To negate the statement “If n is prime, then n is odd or n is 2,” we first need to understand the structure of the original statement. It can be expressed in logical terms as:
p → (q ∨ r)
where:
- p: n is prime
- q: n is odd
- r: n is 2
The negation of an implication, p → (q ∨ r), is given by:
¬(p → (q ∨ r)), which is logically equivalent to p ∧ ¬(q ∨ r).
Now, ¬(q ∨ r) can be simplified using De Morgan’s laws:
¬(q ∨ r) = ¬q ∧ ¬r
So, the complete negation becomes:
p ∧ (¬q ∧ ¬r).
Finally, substituting back we get:
n is prime and n is not odd and n is not 2.
This means the negation states that n is prime, n is even (since not odd), and n is not equal to 2.
In summary, the negation of the original statement is: n is prime and n is even and n is not 2.