Cross section (mathematics)

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In mathematics, the cross section is used to denote certain quantities . If a system of sets is above the base set , then a cross section of is called if all sets intersect in.

The smallest possible thickness of a cross-section of is called the cross-section number of the set system .

Examples

1. Let the number of members of a party with two chairmen be. be the number of interest groups in this party, for example the four groups trade unionists, conservatives, women and medical professionals. Each of these four groups represents a subset of the set of all members in the party. If each of these groups belongs to one of the two chairmen, then they form a cross-section of . For this, for example, the chair would have to consist of a conservative doctor and a trade unionist.
If there is no such couple among all members of the party who together serve all interest groups, the cross-sectional number is greater than 2. If there is a member who is conservative, female and medical at the same time and also belongs to the union, then the cross-sectional number is even only 1.
2. Let be the Euclidean plane and the set of all closed unit squares that intersect the -axis. A cross-section of this system of quantities would be e.g. B. , because every closed unit square that intersects the axis also contains such a point. This cross-section is countably infinite, so the number of cross-sections is at most countably infinite. It is easy to consider that there cannot be a finite cross-section in this system of sets, so countable is infinite.