The lac and trp operons are opposites, and that is the point
Both are prokaryotic operons and both are examples of negative control, but they are regulated in mirror-image ways. The trp operon is repressible: it is normally on, and tryptophan acts as a corepressor that activates the repressor to switch off synthesis of an amino acid the cell no longer needs to make. The lac operon is inducible: it is normally off, and allolactose acts as an inducer that inactivates the repressor so the cell can catabolise lactose when it appears. The generalisation worth carrying is that anabolic pathways tend to be repressible and catabolic pathways inducible, because that is what a cell economising on protein synthesis would do.