For each of the following pairs of statements determine whether the proposed negation is correct. If correct, determine which is true: the original statement or the proposed negation. If the proposed negation is wrong, write a correct version of the negation and then determine whether the original statement or your corrected version of the negation is true.
(a) Statement: For all real numbers x, y, if x2 > y2, then x > y.
Proposed negation: There exist real numbers x, y such that x2 > y2 but x ≤ y.
(b) Statement: There exist real numbers x, y such that x and y are rational but x + y is irrational.
Proposed negation: For all real numbers x, y, if x + y is rational, then each of x, y is rational.
(c) Statement: For all real numbers x, if x is not 0, then x has a multiplicative inverse.
Proposed negation: There exists a nonzero real number that does not have a multiplicative inverse.
(d) Statement: There exist odd integers whose product is odd.
Proposed negation: The product of any two odd integers is odd.