Answer:
The statement [tex](P\lor Q)\rightarrow R[/tex] is logically equivalent to the statement [tex](P\rightarrow R) \land (Q\rightarrow R)[/tex]
Step-by-step explanation:
We have the following statement [tex](P\lor Q)\rightarrow R[/tex] and we need to prove that is logically equivalent to this statement [tex](P\rightarrow R) \land (Q\rightarrow R)[/tex].
To prove the logical equivalence of the statements we are going to use the table of logical equivalences as follows:
[tex](P\lor Q)\rightarrow R \equiv \lnot(P\lor Q)\lor R[/tex] by the logical equivalence involving conditional statement.
[tex]\equiv (\lnot P \land \lnot Q)\lor R[/tex] by de Morgan’s laws.
[tex]\equiv (\lnot P\lor R)\land(\lnot Q \lor R)[/tex] by distributive laws.
[tex]\equiv (P\rightarrow R) \land (Q\rightarrow R)[/tex] by the logical equivalence involving conditional statement.
