from Lecture 7 slide 52 (second bullet), i understand that:

given z from Z_pq the Jacobi symbol is 1 if and only if both z(mod p), z(mod q) are in QR(p), QR(q) or both are not

so from this i conclude that given x from Z_p that is QR(p) and y from Z_q that is QR(q)

if we calculate using the Chinese Remainder, z such that z = x (mod p) and z = y (mod q)

this z is in QR(pq) if and only if both x, y are or both are not..

is this correct so far?

because on slide 57 it says "let's find a z from Z_pq that is NQR(pq) by taking x, y that are NQR(p), NQR(q)"

and it contradicts the first point.. what am i missing?