"He knows and we do not know."
Translation:Він знає, а ми не знаємо.
«І» doesn't work in this sentence, because «і» joins similar things, it marks joined things as belonging to roughly the same group. However, 'he knows' and 'we don't know' is not a similar thing, those sentences are contrasted, so we use «а».
It's all about the agreement between the subjects, verbs and objects of the two sentences you want to connect:
Ти любиш макарони, і ти любиш/ненавидиш рис (the subject is the same, so nothing else matters)
Ти любиш макарони, і вона любить макарони (both love; both love pasta)
Ти любиш макарони, а вона ненавидить (hate) макарони (both are about pasta, but the action is different)
Ти любиш макарони, а вона любить рис (the action is the same, but the object is different)