A problem arises, however, when "both" filters claim to contain the key. We never insert a key into both, so one or both of the filters is lying (producing a false positive), but we don't know which. To determine this, we have another, smaller pair of filters "A"1 and "B"1. "A"1 contains keys that map to 0 and which are false positives in "B"0; "B"1 contains keys that map to 1 and which are false positives in "A"0. But whenever "A"0 and "B"0 both produce positives, at most one of these cases must occur, and so we simply have to determine which if any of the two filters "A"1 and "B"1 contains the key, another instance of our original problem.