Answer Happy

Carefully go through the steps of the proof below. Then answer the questions at the end. Claim. The square of an odd number is odd. Proof. (1) u EZ and u is odd (2) u = 2k + 1 ; by ; by ; by ; by ; by (6) u² ; by (7) u² is odd ; by (8) u Z and u is odd ⇒ u² is odd ; by (9) Vx [x = Z and x is odd ⇒ x² is odd] ; by (1) Decide whether this is: (i) direct proof (ii) proof by contraposition - (iii) proof by contradiction (iv) proof by cases. (2) Decide whether the proof is correct or not. (2a) If the proof is incorrect, identify the first line where a mistake happens. (2b) If the proof is correct, fill in the blanks by choosing from the following list: (i) by algebra - (ii) by assumption - (iii) by closure - (iv) by definition - (v) by definition, for some suitable choice of k (when you enter your answers on Brightspace, you may use the abbreviated form in boldface) - (vi) by logic. [2P] (3) (2k + 1)² (4) u² = 4k² + 4k+1 (5) - u² = 2(2k² + 2k) + 1 2 u arbitrary = 2m + 1, for m = 2k² + 2k