This is a question given as homework. $X = \{(1 0), (0 1), (1, 1)\}$. We have to find all the dichotomies of $X$ and specify which ones are affine and linear separable. In addition, we need to specify the linear and affine separating hyperplane if it exists.
The definition of a dichotomy of $X$ is $(X^+, X_-)$ such that the disjoint union is $X$.
The definition of linear(affine) separable is: A dichotomy in $R^2$ is linear(affine) separable iff there exists $w$ in $R^2$ (and $b$ in $R$) such that $X^+ = \{x \in X\text{ such that }w^T.x(+b) > 0\}$ and $X_- = \{x \in X\text{ such that }w^T.x(+b) <0\}$
I proceeded as follows: Clearly, there are 8 dichotomies. 6 of them are linear separable which can be determined by plotting these points and a line through the origin. Since linear separable implies affine separable, so all 6 are also affine separable.
Two are not linear separable but are affine separable. These are: $X^+ = \{(1 0), (0 1)\}$ and $X_- = \{(1, 1)\}$ and $X^+ = \{(1, 1)\}$ and $X_- = \{(1 0), (0 1)\}$.
My question is: how do I find the linear and affine separating hyperplane (if it exists) for all 8 dichotomies? I think that they should be lines of the form $x = h$ or $y = k$? Won't they depend on the choice of w? Please help me out.