Consider the statement: *'For every positive integer , there is a prime number with .'* Which of the following is its negation?
Answer and explanation
Correct answer: B. There is a positive integer such that no prime number is greater than
- Step 1: The statement has the form .
- Step 2: Negate by swapping each quantifier and negating the inner claim: , i.e. there is an beyond which there are no primes.
- Step 3: (The negation is false, because there are infinitely many primes, but the question only asks for the negation.)
Answer
There is a positive integer such that no prime number is greater than .
Why not the others: 'For every , no prime exceeds ' keeps the wrong quantifier; the options about numbers that are not prime change the inner statement incorrectly; 'every number greater than is prime' is a different, stronger claim.
Practise more Negating statements with quantifiers.
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