Question

Which of the following is the negation of the statement, For all odd primes $p<q$ there exists positive non-primes $r<s$ such that $p^2+q^2=r^2+s^2$.

• A. For all odd primes $p<q$ there exists positive non-primes $r<s$ such that $p^2+q^2=r^2+s^2$.
• B. There exists odd primes $p<q$ such that for all positive non-primes $r<s$, $p^2+q^2=r^2+s^2$.
• C. There exists odd primes $p<q$ such that for all positive non-primes $r<s$, $p^2+q^2\ne r^2+s^2$.
• D. For all odd primes $p<q$ and for all positive non-primes $r<s$, $p^2+q^2\ne r^2+s^2$.

Answer 1

The correct option is C There exists odd primes $p<q$ such that for all positive non-primes $r<s$, $p^2+q^2\ne r^2+s^2$.

For the inverse of a statement, "for all" and "there exists" are interchanged, and "$=$" changes to "$\ne$".

Therefore, the negative of the given statement will be:

There exists odd primes $p<q$ such that for all positive non-primes $r<s$, $p^2+q^2\ne r^2+s^2$