We need the か particle to form “or”. In its simplest form:

Tea or coffee.

For more complicated structures we can think it as a set of embedded questions. We already saw how to embed a question in a sentence here.

I don’t know if the teacher is kind.

All we’re doing now is adding another embedding question using the same grammar.

I don’t know if the teacher is kind or unkind.
I can’t decide whether to buy chocolate or flowers.
I don’t know if she’s a departmental head or a divisional head.
裕輔ゆうすけ 今週末こんしゅうまつは、友達ともだちとキャンプに行ってバーベキューするよ。
Yusuke This weekend I’m going camping with friends and will have a BBQ.
純子じゅんこ そうなの?いいね。
Junko Really? Nice!
裕輔ゆうすけ まあ、それいえでのんびりする。
Yusuke Well, either that or relax at home.

Just like when we looked at the casual form of a question for na-adjectives and nouns, we may see なのか used instead of just か to form the structure.

Let’s ask the waiter whether this is chicken or beef.

Additionally, you may hear それとも used inside the question clause. This does not change the meaning, merely places emphasis on it being either A or B.

I don’t know if that’s a bird or a plane.

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