Already / Still / Not Yet
3 related grammar points that express the same core idea - compared below so you can tell which one to reach for.
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もう / まだ
Someone asks もう食べましたか (have you eaten yet?) and you have not.
まだ~ません vs もう~: Answer with まだ + negative: いいえ、まだ食べていません, or simply まだです. もう + negative means 'not anymore', something that used to happen and has stopped (もう食べません, I won't eat any more), so it does not answer the question. Note the ている: 'not yet' is まだ~ていません, not まだ食べませんでした.
Choosing between 'still' and 'not yet'.
まだ~ません vs まだ〜: The verb decides it. まだ + affirmative is 'still', something continuing (まだ寝ている, still asleep). まだ + negative is 'not yet', something that has not started (まだ起きていない, not up yet).