Question

Use truth tables to show that the following statements are logically equivalent. \(\sim(P \wedge Q \wedge R)=(\sim P) \vee(\sim Q) \vee(\sim R)\)

Short answer

Yes, \(\sim(P \wedge Q \wedge R)\) is logically equivalent to \((\sim P) \vee(\sim Q) \vee(\sim R)\). This is evidenced by their matching truth tables that yield the same truth value for all possible combinations of proposition variable values.

Step-by-step solution

Define the basic truth tables

Define the basic truth tables for the proposition variables \(P, Q, R\). Each of these variables can either be true (T) or false (F), leading to 2^3 or 8 possible combinations.

Calculate the values for \(P \wedge Q \wedge R\)

For each of the 8 combinations defined in the previous step, calculate the value of the expression \(P \wedge Q \wedge R\), which is true only when all three proposition variables are true.

Calculate the values for \(\sim(P \wedge Q \wedge R)\)

Using the truth values of \(P \wedge Q \wedge R\) from step 2, calculate the truth values for \(\sim(P \wedge Q \wedge R)\). This expression negates \(P \wedge Q \wedge R\), and thus is true when \(P \wedge Q \wedge R\) is false, and vice versa.

Calculate the values for \((\sim P) \vee(\sim Q) \vee(\sim R)\)

Now calculate the truth values for \((\sim P) \vee(\sim Q) \vee(\sim R)\), based on the truth values of \(P, Q, R\) defined in step 1. The expression is true when at least one of \(\sim P, \sim Q, \sim R\) is true.

Compare the truth tables

Finally, compare the truth tables for \(\sim(P \wedge Q \wedge R)\) and \((\sim P) \vee(\sim Q) \vee(\sim R)\). If all corresponding truth values match, the two logical expressions are equivalent.