# Combinatorics: Complementary Pair

1. Apr 29, 2015

### Robben

1. The problem statement, all variables and given/known data

My book repeatedly uses the phrase "contains one of each complementary pair of sets" and I am wondering what do they mean by that exactly?

2. Relevant equations

None

3. The attempt at a solution

For example, when it proves that an intersecting family of subsets of $\{1,...,n\}$ satisfies $|F|\le2^{n-1},$ it says the $2^n$ subsets of $X$ can be divided into $2^{n-1}$ complementary pairs $\{A,X \A\}$.

I am not sure what the mean by complementary pairs when referring to an intersecting family.

2. Apr 29, 2015

### Dick

The notation {A,X\A} tells you what they mean by complementary pairs. Each pair consists of a subset A and its complement X\A (the set of all elements of X that aren't in A). The sets A and X\A have empty intersection. So if F is an intersecting family of sets then for each pair you can select at most one of A and X\A to belong to F. If you pick both then F is not intersecting. So the number of elements in F is less than or equal to the number of complementary pairs.

Last edited: Apr 29, 2015
3. Apr 30, 2015

### Robben

Thank you very much for clarifying.