Solution to Consider the statement form (P↓Q)↓R. Now, find a restricted statement form logically equivalent to it, … - Sikademy
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Archangel Macsika

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).

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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.


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