The proteins encoded by all the structural genes of the $lac$ operon are essential for which process?

The regulation of the $lac$ operon by a repressor is known as:

Select two statements out of the four given statements $(i-iv)$ with respect to the $lac$ operon.
$(i)$ Glucose or galactose binds with the repressor and inactivates it.
$(ii)$ In the absence of lactose,the repressor binds with the operator.
$(iii)$ The $Z$-gene codes for permease.
$(iv)$ It was elucidated by Francois Jacob and Jacques Monod.

Which of the following does not act as an inducer in the $lac$ operon?

Which of the following is the substrate for $\beta$-galactosidase?

What does the $i$ gene encode in the $lac$ operon?