Question

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

Short answer

Based on the truth table created, it is determined that the two logical statements \(\sim(P \vee Q \vee R)\) and \((\sim P) \wedge(\sim Q) \wedge(\sim R)\) are logically equivalent.

Step-by-step solution

Construct Base Truth Table

Firstly, construct a base truth table with all possible combinations of truth values for P, Q and R. There will be 8 combinations, as there are three variables, and two possible states (true or false) for each variable.

Evaluate Individual Statements

Now, on this base truth table, evaluate the two logical statements separately. For the first statement, \(\sim(P \vee Q \vee R)\), apply the OR operator to P, Q and R first, then negate the result. For the second statement, \((\sim P) \wedge(\sim Q) \wedge(\sim R)\), apply the NOT operator to P, Q, and R individually to get \(\sim P\), \(\sim Q\), \(\sim R\) respectively, then apply the AND operator to these three results.

Compare the Results

Finally, compare the truth values calculated for \(\sim(P \vee Q \vee R)\) and \((\sim P) \wedge(\sim Q) \wedge(\sim R)\) in each row of the truth table. If these values are same for each row, then the two logical statements are equivalent. If there are any discrepancies, they are not.

Key concepts

Truth Tables

Truth tables are a fundamental tool in propositional logic, used to determine the validity of expressions involving logical operators. They systematically list all possible combinations of truth values for a given set of propositions and then compute the corresponding outputs for complex logical statements.

For example, in the exercise provided, the truth table starts by listing all 8 possible truth combinations for the propositions P, Q, and R. Once this base is laid out, truth tables allow for the exploration of the relationships between different logical operators by calculating their outcome for each of these combinations. This provides a clear and organized way to ascertain the logical equivalence of two expressions, such as the given \(\sim(P \vee Q \vee R)\) and \(\sim P) \wedge(\sim Q) \wedge(\sim R)\).

Using truth tables not only helps in understanding logical equivalence but also aids in visualizing how each operator affects the overall truth value of compound statements, thus reinforcing the grasp of logic fundamentals in students.

Logic Operators

Logic operators, also known as logical connectives, are symbols or words used to connect two or more propositions in propositional logic to form a more complex statement. Some of the basic logical operators include:

• \(\land\) (AND): It yields true if both connected propositions are true.
• \(\lor\) (OR): It yields true if at least one of the connected propositions is true.
• \(\sim\) (NOT): It negates the truth value of the proposition it precedes – from true to false, or false to true.

When solving logical problems, understanding how each operator affects the truth value of propositions is crucial. In the exercise, we see the application of NOT to each of P, Q, and R individually, and then applying AND to the negated values. Logic operators act as the building blocks for formulating and analyzing logic statements and mastering their use is essential for understanding more complicated logical expressions.

Propositional Logic

Propositional logic is a branch of logic that deals with propositions and their relations and functions in truth-functional contexts. A proposition is a statement that is either true or false, such as 'It is raining,' which has a clear truth value. In computational logic and mathematics, a proposition can be represented by a variable like P, Q, or R.

In propositional logic, complex expressions are built using simple statements and logical operators. These expressions can then be analyzed to determine their truth values in various scenarios. The exercise provided showcases an instance where two logical expressions are tested for logical equivalence, a core concept within propositional logic. Here, the goal is to determine whether \(\sim(P \vee Q \vee R)\) is equivalent to \(\sim P) \wedge(\sim Q) \wedge(\sim R)\), illustrating an application of propositional logic's systematic and formal approach to understanding the structure of arguments and proving propositions.