Below are five statements labelled (A)–(E) that together establish that the sum of two odd integers is even. The steps are not in the correct logical order.
(A) x+y=(2k+1)+(2l+1)
(B) Since x and y are odd integers, there exist integers k and l such that x=2k+1 and y=2l+1.
(C) x+y=2(k+l+1), which is divisible by 2.
(D) Therefore x+y is an even integer.
(E) An integer of the form 2m+1 is odd for some integer m.
Answer the following:
a. (i) Arrange the statements (A)–(E) in a logical sequence to form a valid proof.
b. (ii) Rewrite the complete proof in paragraph style, clearly marking the sections ‘Given’, ‘To Prove’, ‘Proof’, and ‘Conclusion’.
c. (iii) For each step in your ordered proof, name the algebraic property, definition, or prior fact that justifies the step.