Answer to Question #316040 in Discrete Mathematics for Pravin

Question #316040

1.  Prove or disprove that the two propositions in each pair are equivalent.

a)   (p \to (q \to r)) , ((p \to q ) \to ( p \to r ))


1
Expert's answer
2022-03-28T06:14:42-0400

Remind that logical implication \rightarrow can be rewritten via a logical disjunction. Namely, ab=aˉba\rightarrow b=\bar{a}\lor b. Thus, the first statement can be rewritten as: p(qr)=pˉqˉrp\rightarrow(q\rightarrow r)=\bar{p}\lor{\bar{q}}\lor r. The second statement can be rewritten as: (pq)(pr)=(pˉq)(pˉr)(p\rightarrow q)\rightarrow(p\rightarrow r)=\overline{(\bar{p}\lor q)}\lor(\bar{p}\lor r). Remind that a negation of a disjunction can be rewritten via conjunction. Namely, (pˉq)=pqˉ\overline{(\bar{p}\lor q)}=p\land {\bar{q}}. It can checked via the truth table. Thus, we get: (pq)(pr)=(pqˉ)(pˉr)(p\rightarrow q)\rightarrow(p\rightarrow r)=(p\land {\bar{q}})\lor(\bar{p}\lor r). To complete the proof, it is enough to check that (pqˉ)pˉ=pˉqˉ(p\land {\bar{q}})\lor{\bar{p}}=\bar{p}\lor{\bar{q}} . It holds, since the statement ppˉp\lor{\bar{p}} is tautology, From the latter we get: (pqˉ)(pˉr)=pˉ(qˉr)(p\land {\bar{q}})\lor({\bar{p}}\lor r)=\bar{p}\lor({\bar{q}}\lor r). Thus, both statements are equivalent.


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