This sentential logic problem is stated as: Suppose that $A \models B$, where $A$ is a conjunction of literals and $B$ is a disjunction of literals. Show that $ \models \neg A$, $ \models B$, or a conjunct of $A$ is a disjunct of $B$.

What do the terms "conjunct of $A$" and "disjunct of $B$" mean?


1 Answer 1


If $A = p \land q$ (so $A$ is a conjunction of literals), and $B = r\lor s$ (so $B$ is a disjunct of literals),

then $p, q$ are each conjuncts of $A$ and $r, s$ are each disjuncts of $B$.

The same would be true for arbitrarily number of literals involved:

If $A = P_1 \land P_2 \land \cdots \land P_n,\;$ then any literal $\,P_i$ is a conjunct of $A\;$ (where $1 \leq i \leq n$).

If $B = Q_1 \lor Q_2 \lor \cdots \lor Q_m,\;$ then any literal $\,Q_j$ is a disjunct of $B\;$ (where $1\leq j\leq m$).

You are being asked to show that, e.g., there is some $\,i,\,j,\,$ such that $\,P_i = Q_j$.


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