Section $4\mathrm{.}2$

Prove that if a and $b$ are any odd integers, then $a^2+b^2$ is even.

Prove that the difference between the squares of any two consecutive integers is odd.

Determine whether the statement is true or false. Justify your answer with a proof or a counterexample, as

appropriate. In each case use only the definitions of the terms and the Assumptions listed, not any previously

established properties.

i$.$ For all integers a and $b,4a+5b+3$ is even.

ii$.$ The difference of any two odd integers is even.