I(C)$, so $x_1 + x_2 + x_3 + x_4 \notin J_C \subseteq I(C)$. This shows that the first implication is false.

As before, however, we can also show that the second implication is false. The code $C' = \{\emptyset, 12, 13, 14, 123, 145\}$ again serves as a counterexample. By the proof of Lemma 3.1, $x_1 + x_2 + x_3 + x_4+x_5 \notin I(C')$, so $x_1 + x_2 + x_3 + x_4+x_5 \notin J_{C'} \subseteq I(C')$.

[*Converse of Type 6:*]{} Finally, the proof of Lemma 3.1 tells us that for the code $C = \{1,2,12\}$ pictured in Figure \[fig:T6C\], $x_1+x_2+1 \notin I(C)$. Therefore, $x_1+x_2+1 \notin J_C \subseteq I(C)$.

Notice that if we replace $J_C$ with $I(C)$ in the Type 4 relation, the counterexample we used in the proof of Theorem 3.2 no longer works since $x_1(1+x_1) + x_2(1+x_2) \in I(C)$. In fact, we will prove in the next section that the converse of the Type 4 relation is actually true when $J_C$ replaced with $I(C)$. This suggests that the Type 4-6 relations may be modified so that their converses hold. Possible modifications will be discussed in Section 4.

We finish this section by showing that the counterexample codes presented all 