Statement I: The z gene in the lac operon encodes β-galactosidase, an enzyme that breaks down lactose into galactose and glucose. This statement is correct.
Statement II: Lactose (or its isomer, allolactose) induces the lac operon. Glucose, conversely, represses it, a process known as catabolite repression, even when lactose is present. Galactose does not directly induce the lac operon; the induction is dependent on lactose. Consequently, statement II is incorrect.