Version Spaces

We will try to output a description of the set of all hypothesis consistent with the set of training examples.
A hypothesis $h$ is consistent with a set of training examples $D$ of target concept $c$ if and only if $h(x)=c(x)$ for each training example $\langle x, c(x) \rangle$ in $D$. \[Consistent(h,D) \equiv \forall_{\langle x, c(x) \rangle \in D} h(x)=c(x) \]
Notice how an example $x$ satisfies $h$ when $h(x)=1$, while it is consistent with $h$ if $h(x)=c(x)$.
The version space, denoted $VS_{H,D}$ with respect to $H$ and $D$, is the subset of hypothesis from $H$ consistent with the training examples in D. \[VS_{H,D} \equiv \{h \in H\,|\, Consistent(h,D)\}\]