Structure of a Korean maths question (단, 대하여)

Question

While looking at some 수능 maths questions, I noticed some common pattern using the words 단 and 대하여. Here is an example:

전체집합 U={1,2,3,4,5,6}에 대하여 두 부분집합 A, B는 다음 조건을 만족시킨다. 두 지합 A화 B를 정하는 경우의 수는? (단, n(X)는 집합 X의 원수의 개수이다.)

(가) $A\cap B=\emptyset$

(나) n(A)=1, n(B)>=1

Taking a lot of liberty with the structure of the sentence, this can be translated as

A and B are two subsets of the set U={1,2,3,4,5,6} that satisfy the 2 conditions below. In how many ways can the sets A and B be chosen? (Given, n(X) is the number of elements of the set X)

Condition 1. $A\cap B=\emptyset$

Condition 2. $n(A)=1, n(B)>=1$

My (related) questions:

1. What are the words 단 and 대하여?
2. How are they usually translated in English?
3. How are they used in a non-mathematical context?

Answer 1

대하여 is a very common word, meaning "concerning/on/with respect to". It is actually derived from verb 대하다 (although it's probably best regarded as a separate word), and can take forms as 대하여/대해/대한.

일본에 대하여 알아보자. = Let's know more about Japan.

영문법에 대한 책 = A book on English grammar

전체집합 U에 대하여... = With respect to the universal set U...

단 is a connecting word used to quantify the previous sentence or explain an exception, similar to "Except that" or "Note that".

수수료는 500원입니다. 단, 주말에는 추가 요금이 부과될 수 있습니다. = The commission is 500 won, except that weekend surcharge may apply.

X의 값을 구하라. 단, X는 양수이다. = Find the value of X. Assume that X is positive.