Question

Consider the statement form (P↓Q)↓R.

Now, find a restricted statement form logically equivalent to it, in

a) Disjunctive normal form (DNF).

b) Conjunctive normal form (CNF).

Answer 1

Let us find a restricted statement form logically equivalent to $(P \downarrow Q) \downarrow R$ , in

a) Disjunctive normal form (DNF).

Since

$(P \downarrow Q) \downarrow R\equiv \neg(\neg(P\lor Q)\lor R) \equiv \neg(\neg(P\lor Q))\land \neg R \\\equiv (P\lor Q)\land \neg R \equiv (P\land \neg R)\lor( Q\land \neg R),$

we conclude that $(P\land \neg R)\lor( Q\land \neg R)$ is a disjunctive normal form of a restricted statement.

b) Conjunctive normal form (CNF).

Taking into account that

$(P \downarrow Q) \downarrow R\equiv \neg(\neg(P\lor Q)\lor R) \equiv \neg(\neg(P\lor Q))\land \neg R \equiv (P\lor Q)\land \neg R,$

we conclude that $(P\lor Q)\land \neg R$ is a conjunctive normal form of a restricted statement.