What is the meaning of Quranic verses such as: "that Allah may know those of you who believe," or "that Allah may know those of you who strive," and do they contradict the fact that Allah is all-knowing of everything?

The verses indicate that the knowledge of Allah mentioned therein is the knowledge of manifestation and revelation, not a new knowledge for Him. He has written and known everything before its existence, and He merely manifests it at its appointed time to establish proof against His servants. Allah knows a thing before it comes into being; He is the Knower of the Unseen, and then He becomes the Knower of the witnessed after its existence. His eternal, singular knowledge does not change with the changing states of the known things. However, the knowledge upon which reward and punishment are based is the knowledge of manifestation after the act has occurred and people have been tested.