Beginning Activity Beginning Activity 1: Using a Logical Equivalency
1.
Complete a truth table to show that \(\left( {P \vee Q} \right) \to R\) is logically equivalent to \(\left( {P \to R} \right) \wedge \left( {Q \to R} \right)\text{.}\)
2.
Suppose that you are trying to prove a statement that is written in the form \(\left( {P \vee Q} \right) \to R\text{.}\) Explain why you can complete this proof by writing separate and independent proofs of \(P \to R\) and \(Q \to R\text{.}\)
3.
Now consider the following proposition: For all integers \(x\) and \(y\text{,}\) if \(xy\) is odd, then \(x\) is odd and \(y\) is odd.Proposition.
4.
Now prove that if \(x\) is an even integer, then \(xy\) is an even integer. Also, prove that if \(y\) is an even integer, then \(xy\) is an even integer.
5.
Use the results proved in Exercise 4 and the explanation in Exercise 2 to explain why we have proved the contrapositive of the proposition in Exercise 3.