Linear separability

In geometry, two sets of points in a two-dimensional space are linearly separable if they can be completely separated by a single line. In general, two point sets are linearly separable in n-dimensional space if they can be separated by a hyperplane.

In more mathematical terms: Let X₀ and X₁ be two sets of points in an n-dimensional space. Then X₀ and X₁ are linearly separable if there exists n+1 real numbers w₁,w₂,..,w_{n + 1}, such that every point $x \in X_{0}$ satisfies $\sum^{n}_{i=1} w_{i}x_{i}\ge w_{n+1}$ and every point $x \in X_{1}$ satisfies $\sum^{n}_{i=1} w_{i}x_{i} < w_{n+1}$, where x_i is the i:th component of x

Number of linearly separable Boolean hypercubes in each dimension (sequence A000609 in OEIS)

Dimension	Linearly separable Boolean hypercubes
2	14
3	104
4	1882
5	94572
6	15028134
7	8378070864
8	17561539552946
9	144130531453121108

Usage

Linear separability allows simple Classification in machine learning.