(1)
{p, q, pq} is linearly independent as P
deg p ≥ 1 and deg q ≥ 1 as Q
~P is {p, q, pq} is linearly dependent
~Q is p < 1 or deg q < 1
Then ~Q => ~P is:
~Q is p < 1 or deg q < 1 =>~P is {p, q, pq} is linearly dependent
PD: Is contraposition the right word? I searched in wiki to know how it's named in English, in Spanish is "contrarecripoco" and I thought it translation was "transposition"
(2) I thought something like:
If deg p ≥ 1 and deg q ≥ 1, then deg (pq) ≥ 2, so λ_{1}p + λ_{2}q + λ_{3}pq = 0 only if λ_{3}=0, and as p and q are linearly independent...
I don't remember exactly how to show that degres must be lesser than 1, but if you write both polynomials as sums, then the product of them will have a x with a power greater than one and it thought it was trivial enough to say that you can't make λ_{1}*(0)+λ_{2}*(0)+λ_{3}*aX^{n}=0 (with a≠0) without making λ_{3}=0 (because the polynomial sum to be equal to zero needs to be every coefficient of a power of X be equal to zero).