Lactose is a combination of one galactose and glucose molecule.

E. coli can degrade lactose into galactose and glucose by LacZ.

If the cells are in the glucose-rich condition, the pBAD promoter activity is repressed.

Then, if we induce pBAD promoter in plasmid by arabinose when the cells are in the lactose-rich condition, will transcription be repressed?

If the answer to this question is yes, what kind of inducible promoter can I use in lactose rich condition? Lactose inducible promoter is not allowed because of my experimental design.