3. Consider the following statement and ‘proof.’ Statement. Let x, y ∈ Z. If x3 − y2 is odd, then either x is even and y is odd, or x is odd and y is even. Proof. Suppose x, y are integers, and suppose that it is not true that either x is even and y is odd, or x is odd and y is even. We will show x3 − y2 is even. Since it is not true that either x is even and y is odd, or x is odd and y is even, then we can assumethat both x and y are odd. Since x and y are odd, by definition, x = 2k + 1 and y = 2L + 1 for some k,L ∈ Z. Observe x3 − y2 = (2k + 1)3 − (2L + 1)2 = (8k3 + 12k2 + 6k + 1) − (4L2 + 4L
3. Consider the following statement and ‘proof.’
Statement. Let x, y ∈ Z. If x3 − y2 is odd, then either x is even and y is odd, or x is odd and y is even.
Proof. Suppose x, y are integers, and suppose that it is not true that either x is even and y is odd, or x is odd and y is even. We will show x3 − y2 is even.
Since it is not true that either x is even and y is odd, or x is odd and y is even, then we can assumethat both x and y are odd. Since x and y are odd, by definition, x = 2k + 1 and y = 2L + 1 for some k,L ∈ Z. Observe
x3 − y2 = (2k + 1)3 − (2L + 1)2
= (8k3 + 12k2 + 6k + 1) − (4L2 + 4L + 1)
= 8k3 + 12k2 + 6k − 4L2 − 4L
= 2(4k3 + 6k2 + 3k − 2L2 − 2L)
Since k, L are integers, (4k3 + 6k2 + 3k − 2L2 − 2L) is an integer. So by definition, x3 − y2 is even.
(a) What kind of proof is being attempted? How do you know?
(b) There is something wrong with this proof.
What is it? Explain.
(c) Correct the proof.
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