Combinatorics: Complementary Pair

[Robben]

Homework Statement

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?

Homework Equations

None

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.

 

[Dick]

Homework Statement

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?

Homework Equations

None

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.

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.

 

[Robben]

Thank you very much for clarifying.