Question

Use logical cquivalences or a truth table to show that (PVQ) R to (P R) A (Q R)

Answer 1

The statement
`(P\lor Q)\rightarrow R` is logically equivalent to the statement
`(P\rightarrow R) \land (Q\rightarrow R)`

Explanation:

We have the following statement
`(P\lor Q)\rightarrow R` and we need to prove that is logically equivalent to this statement
`(P\rightarrow R) \land (Q\rightarrow R)`.

To prove the logical equivalence of the statements we are going to use the table of logical equivalences as follows:

`(P\lor Q)\rightarrow R \equiv \lnot(P\lor Q)\lor R` by the logical equivalence involving conditional statement.

`\equiv (\lnot P \land \lnot Q)\lor R` by de Morgan’s laws.

`\equiv (\lnot P\lor R)\land(\lnot Q \lor R)` by distributive laws.

`\equiv (P\rightarrow R) \land (Q\rightarrow R)` by the logical equivalence involving conditional statement.