Nashi (なし)
meaning: without
問題なし - mondai-nashi (without problem) 異常なし 文句無し - monku-nashi (undisputed, entirely, perfect)
Example
休日なしで働く。 - kyuujitsu-nasi de hataraku (work without holidays)
Like ない, なし means not exist, not present, without, missing. We can see an example where the meaning of ない and なし is the same:
mondainai. == mondainashi.
Note
Although なし is an adjective, it is considered a noun, and used as such.
We can characterize the use of なし according to the following guidelines:
なし is commonly linked to other nouns without using any particle, while it is more common to use a particle with ない (問題がない)
Since なし is considered a noun, it can be followed by the copula だ, by the particle の, and by に to become adverb. In this sense, なし has the same meaning and usage as ないこと
なし can be translated as not or without depending on the case
なしで - Without (most common usage)
Example
あなたなしで生き[ikiru]ていけない。
I can't live without you.
In this sentence, the particle で indicates that without (なし) you (あなた) I cannot live (生きていけない).
なしに
We can also find the particle に instead of で. なしに (nashi ni) takes on the same meaning as なしで: without.
Examples
今日[kyou]は昼飯[hirumeshi]なしにする。
I decided to go without lunch today.
Note: nashi ni suru (to going with, to decided)
Examples
子育て[kosodate]中[naka]の車[kuruma]なし生活[seikatsu]
Living without a car while raising children
seikatsu - living, life
WTF: 車なし子育て歴5年のママの意見も紹介
kurumanashi kosodate reki 5 nen no mama no iken mo shoukai
It introduces also some opinions from mothers who have been raising children without a car for 5 years.
Igai
meaning: except, apart from, exception of
Examples
食事以外には何ができる?
shokuji igai ni wa nani ga dekiru?
Apart from eating, what else can we do?
彼女以外誰もいない。
kanojo igai dare mo inai.
There's no one else but her.
水以外何も見えない。
mizu igai nanimo mienai.
I don't see anything other than water.
彼は英語以外の言語は全く知らない。
kare wa eigo igai no gengo wa mattaku shiranai.
He knows no languages other than English.