disjunct set D is tautological $\iff$ D contains both p and $\neg p$ We say that a set of literals D has model the interpretation I, if which there's a literal p in D, s.t. $I \models p$. D is tautological if every interpretation is a model for D. How do you prove that a set of literals D (disjunct set) is tautological if and only if it contains both literals p and $\neg p$?

For every interpretation $I$ exactly one of $p$ and $¬p$ is evaluated to TRUE.

Thus, for every $I$ we have that $I \vDash D$, and this means that $D$ is tautological.

For the vice versa, assume that $D$ is tautological and that there is no atom negated and unnegated.

If so, we can define an interpretation $I$ as follows:

> if $p∈D$, then $I(p)=False$ and if $¬p∈D$, then $I(p)=True$.

We have $I \
vDash D$, contradicting the fact that $D$ is tautological.