Question

The negation of the statement “For all real numbers x and y, x + y = y + x” is

• A. for all real numbers x and y, x + y ≠ y + x
• B. for some real numbers x and y, x + y = y + x
• C. for some real numbers x and y, x + y ≠ y + x
• D. for some real numbers x and y, x - y = y - x

Answer 1

The Correct Option is C

The given statement is "For all real numbers x and y, x + y = y + x".

The negation of "For all" is "There exists some" or "For some".

The negation of "=" is "\(\neq\)".

Therefore, the negation of the statement is: "For some real numbers x and y, x + y \(\neq\) y + x".

Thus, the correct option is (C) for some real numbers x and y, x + y \(\neq\) y + x.

Answer 2

The given statement is:
"For all real numbers xx and \(yy, x+y=y+xx + y = y + x\)".

To negate this, we need to change the universal quantifier ("For all") to an existential quantifier ("For some") and reverse the equality.

The negation would be:
"There exist some real numbers xx and yy such that \(x+y≠y+xx + y \neq y + x\)".

Answer:
The correct option is (C):
For some real numbers xx and \(yy, x+y≠y+xx + y \neq y + x.\)