Which of the following is a negation for "For all real numbers r , there exists a number s such that $r s 10 . "$
A) There exists a real number r such that for all real numbers s, $r s \ngtr 10 .$
B) For all real numbers r , there does not exist a number s such that $r s 10 .$
C) There exists real numbers r and s such that $r s \ngtr 10 .$
D) For all real numbers r and $s , r s \ngtr 10 .$
E) There exists a real number r and there does not exist a real number s such that $r s \ngtr 10$
F) For all real numbers r , there exists a number s such that $r s \ngtr 10 .$
G) There exists a real number r such that there does not exist a real number s with $r s \neq 10$

Question

Which of the following is a negation for "For all real numbers  r , there exists a number  s  such that $r s > 10 . "$
A) There exists a real number  r  such that for all real numbers s, $r s \ngtr 10 .$
B) For all real numbers  r , there does not exist a number  s  such that $r s > 10 .$
C) There exists real numbers  r  and  s  such that $r s \ngtr 10 .$
D) For all real numbers  r  and $s , r s \ngtr 10 .$
E) There exists a real number  r  and there does not exist a real number  s  such that $r s \ngtr 10$
F) For all real numbers  r , there exists a number  s  such that $r s \ngtr 10 .$
G) There exists a real number  r  such that there does not exist a real number  s  with $r s \neq 10$

Quizplus · Accepted Answer

The correct negation of "For all real numbers $r$, there exists a number $s$ such that $rs > 10$" is "There exists a real number $r$ such that for all real numbers $s$, $rs \ngtr 10$." This is because the negation involves switching "for all" to "there exists" and "there exists" to "for all," and changing $>$ to $\ngtr$ (not greater than), which effectively means $rs \leq 10$.

Which of the Following Is a Negation for "For All rs>10."r s > 10 . "rs>10."

Which of the Following Is a Negation for "For All $r s > 10 . "$