rectangle (4,2); (-2,-2) rectangle (0,2); (2,-2) rectangle (4,2); (X) at (3.6,1.6) [$X$]{}; (L1) at (-1,0) [$U_1$]{}; (L2) at (1,0) [$U_1 \cap U_2$]{}; (L3) at (3,0) [$U_2$]{};

; ; ; ; ;

node\[left\] [$U_{2}$]{}; node\[right\] [$U_{3}$]{}; node\[above\] [$U_4$]{}; node [$U_5$]{}; (-2,-2) rectangle (4,2); ; (X) at (3.6,1.6) [$X$]{}; (Y) at (-.4,-2.5) [$U_1$]{};

For each of the following relations, there exists a collection of sets $\mathcal{U} = \{U_i\}^n_{i=1}$ in a stimulus space $X$ (with corresponding neural code $C = C(\mathcal{U})$) and $m \geq 2$ indices $1 \leq i_i < i_2 < ... < i_m \leq n$ such that the relation does [**not**]{} hold:

-   1.  $m$ is even and $U_{i_k} \subseteq \cup_{j \in [m] \setminus  \{k\}} U_{i_j}$ for all $k = 1,...,m \Rightarrow x_{i_1} + ... + x_{i_m} \in I(C)$, and

    2.  $m$ is odd, $U_{i_k} \subseteq \cup_{j \in [m] \setminus  \{k\}} U_{i_j}$ for all $k = 1,...,m$, and $\cap_{k=1}^m U_{i_k} = \emptyset \Rightarrow x_{i_1} + ... + x_{i_m} \in I(C)$.

-   $\cup^m_{k=1} U_{i_k} = X \Rightarrow  x_{i_1} + ... + x_{i_m} + 1 \in I(C)$

[*Converse of Type 5:*]{} To disprove the converse of the Type 5 relation, it is enough to prove that the first implication is false. To this 