Smash

In mathematics, smashing is a concept from the areas of group theory and set theory .

definition

Be a system of sets and . is - shattered , d. H. if and only if you can create any subset of by intersecting an element of with . ${\displaystyle (X,{\mathcal {F}})}$ ${\displaystyle A\subseteq X}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle :\Leftrightarrow \{F\cap A|F\in {\mathcal {F}}\}={\mathcal {P}}(A)}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle A}$

example

Be , . Now , it is of shattered, since one each of the subsets of the three points can separate by means of a closed half-plane. ${\displaystyle X:=\mathbb {R} ^{2}}$ ${\displaystyle {\mathcal {F}}:=\{{\text{alle abgeschlossenen Halbebenen im }}\mathbb {R} ^{2}\}}$ ${\displaystyle A=\{{\text{drei Punkte im }}\mathbb {R} ^{2},{\text{ die nicht auf einer Gerade liegen}}\}}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {F}}}$

If, on the other hand, the points are all on a straight line, the middle point cannot be separated from the other two and thus is not shattered by. Furthermore, no 4 points in the plane can be shattered by closed half-planes. ${\displaystyle A}$ ${\displaystyle {\mathcal {F}}}$

Web links

• Eric W. Weisstein : Shattered Set . In: MathWorld (English).