Linear separators in the plane — VC = 3

Parent: examples Prereq: learning-theory — H, h, shattering; ℝ.

Class $H = \{\, h_{w,b} : w \in \mathbb{R}^2,\ b \in \mathbb{R} \,\}$ (weight vector w, offset b):

A single straight line splits the plane into +/− (one hypothesis = one cut). It shatters any 3 non-collinear points — all 2³ = 8 labelings — but fails on 4 in the XOR arrangement: the two + corners lie on one diagonal, the two − on the other, and the diagonals cross at the center, so no single line can put both + on one side and both − on the other.

Yes, two lines could carve out XOR — but "two lines" is a different, higher-VC class. VC counts what one hypothesis from your class can do; for linear separators that's one straight cut.

(In general, linear separators — homogeneous halfspaces plus offset — in ℝᵈ have VC = d+1.)