Absence and nothingness with

means a state where an object or action lacks purpose, value, or effect. It expresses that no matter how hard you try, it will not lead to results, or that something makes no sense (nonsense).

Ability and Knowledge

(Incompetent)

無能(むのう) means incompetent or incapable. It can describe a person or the lack of ability itself.

It combines 無 (nothing) with (ability) to show lack of ability.

Manga panel from 名探偵コナン showing example of An incompetent officer with 無能.
名探偵コナン » Volume 7 » Page 8

Police detective Yokomizo isn’t about to let Sasai, his prime murder suspect, leave. Sasai reminds the officer that he has photographic proof that he wasn’t near the victim at the time of the murder. He calls for Detective Mouri to back him up.

笹井(ささい):
名探偵(めいたんてい)のあなたからもいってやってくださいよ。 この刑事(けいじ)がいかに無能(むのう)かという(こと)を…」
“You, the great detective, should tell him yourself. Just how incompetent this police detective is...”
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Limits and Possibility

(Impossible)

無理(むり) means impossible, unreasonable, or too demanding. In conversation, it can mean “that’s impossible” or warn someone not to push too hard.

It combines 無 (without) with 理 (reason) to show something without reasoning.

Manga panel from ホリミヤ showing example of I'd totally die in the heat with 〜じゃう.
ホリミヤ » Volume 1 » Page 45

On a summer day, Hori and her friends in class can’t ignore the heat.

由紀(ゆき):
(そと)(あつ)さってなんなの…? クーラーないと()んじゃうって…っ」
“What's with this heat outside...? I'd totally die without the AC...”
「この日差(ひざ)しのなか体育(たいいく)とか絶対(ぜったい)無理(むり)だわ…」
“There's no way I can do PE in this blazing sun...”
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Presence and Omission

(Safety)

無事(ぶじ) means safe, unharmed, or without incident.

It combines 無 (without) with (incident) to show being without incident.

Manga panel from ストロボ・エッジ showing example of Furthermore, it's just a bonus item with それに〜.
ストロボ・エッジ » Volume 1 » Page 23

After school, Ninako notices Ren sleeping a few seats away on the train. When Ren wakes and rushes for his stop, he knocks Ninako’s phone from her hand. The phone hits the floor, and the strap charm breaks. Ren misses his exit while apologizing and offers to replace the charm.

仁奈子(になこ):
「だっ 大丈夫(だいじょうぶ)ですっ」
“I—it's all right.”
「ケータイは無事(ぶじ)だし!」
“Since my cell phone is safe!”
「それにこのストラップただのジュースのおまけだし」
“Furthermore, this strap is just a juice prize.”
「しかももうだいぶ(まえ)〜〜〜〜のやつで(ふる)いから」
“Plus, it's from way back when, so it's old.”
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