what is the difference between だめな and むりな　?

Question

Answer

**** Etymological explanation

- 無理： 理が無い（ので達成できない）
you can't acieve it because it is unreasonable

- 駄目： it makes no difference (やっても無意味、無価値）
This word originates from GO (囲碁) game, 
駄目 in 囲碁　means "useless/worthless slot", which turns to mean simply "useless/worthless".

＊＊＊＊＊

However, nowadays 駄目 is often used to express prohibition, "Don't do that"

「だめ、来ないで」
「薬物濫用。ダメ！ゼッタイ　http://www.dapc.or.jp/」

you usually use むだ to express "worthless/useless even if you try".

＊＊＊＊ example
このプールは深いから、子供が泳ぐのは無理だ。: A description with a reason.
Children can't swim here because they are small but the pool is deep.
このプールは深いから、子供は泳いじゃダメ : A prohibition
Children is not allowed to swim here because it is dangerous.

Answer

無理な ≒ impossible/unreasonable
駄目な ≒ bad/useless

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