I have been reading sieve theory from notes of zeev rudnick here :http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and in page 2 of lecture 16(http://www.math.tau.ac.il/~rudnick/courses/sieves2015/arithmetic%20large%20sieve%20intro.pdf) I have 2 questions.
Question 1: I don't understand how the author wrote $\square [X] \subseteq S(A, P, \Omega)$.
Question 2 : I am not able to deduce how #$ \Omega (p) = (p-1)/2$.
In question 2, I have to prove that either number of non-squares mod p are equal to (p-1)/2 or number of squares mod p are ( p+1)/2 but I am not sure which result to use to prove any of that.
Please help.
For question $2$, notice that $0$ is a perfect square. Now there are $p-1$ numbers remaining, and both $n^2$ and $(-n)^2$ get sent to the same number but are distinct (note $p>2$ for them to be distinct). Second degree polynomials can only have two roots in a (finite) field, so exactly two numbers are mapped to each perfect square in this manner. This shows there are $\frac{p-1}{2}$ total squares not including $0$, which gives $\frac{p+1}{2}$ in total.
For question $1$, $\square [X]$ is the set of square numbers up to some value. Certainly a perfect square is also a quadratic residue for each prime. The sieving process here is going to each prime, and removing all numbers that are not quadratic residues. So perfect squares are never removed. This shows every perfect square up to a given value is in $S(A, P, \Omega)$, which gives the subset relation.