Consider the following theorem: “If xy = 0, then x = 0 or y = 0.” Indicate what, if anything, is wrong with each of the following “proofs.” (a) Suppose xy = 0 and x ≠ 0. Then dividing both sides of...

Consider the following theorem: “If xy = 0, then x = 0 or y = 0.” Indicate what, if anything, is wrong with each of the following “proofs.”

(a) Suppose xy = 0 and x ≠ 0. Then dividing both sides of the first equation by x we have y = 0. Thus if xy = 0, then x = 0 or y = 0.

(b) There are two cases to consider. First suppose that x = 0. Then x ⋅ y =0 ⋅ y = 0. Similarly, suppose that y = 0. Then x ⋅ y = x ⋅ 0 = 0. In either case, x ⋅ y = 0. Thus if xy = 0, then x = 0 or y = 0.