Question

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

Short answer

Upon performing the logical operations for each side of the equation for every possible combination of P, Q, and R, it can be verified that the outcomes are identical for all rows, thus confirming that the two logical expressions, P \wedge(Q \vee R) and (P \wedge Q) \vee(P \wedge R), are indeed logically equivalent. This proves the Distributive Law in propositional logic.

Step-by-step solution

Creating the Truth Table

Begin by creating a truth table. There are three variables, P, Q, and R. This means there will be \(2^3 = 8\) rows in the truth table to cover all the possible combinations of P, Q, and R. For each row, calculate the outcome of both sides of the logical equation P \wedge(Q \vee R) and (P \wedge Q) \vee(P \wedge R).

Fill in the Truth Table for P \wedge(Q \vee R)

Now, for the left side of the equation P \wedge(Q \vee R), calculate the result for each row in the table. To do this, take the logical AND of P with the logical OR of Q and R.

Fill in the Truth Table for (P \wedge Q) \vee(P \wedge R)

Then, for the right side of the equation (P \wedge Q) \vee(P \wedge R), again calculate the result for each row in the table. This time, take the logical OR of the logical AND of P and Q with the logical AND of P and R.

Compare the Outcomes

Finally, compare the outcomes of both expressions for each row. If the outcomes are identical for all rows, it means that the two expressions are logically equivalent.

Key concepts

Logical Conjunction

Imagine you're planning a picnic and want both sunny weather and a free park to enjoy your day. In propositional logic, this 'and' scenario is known as a logical conjunction. It represents a situation where multiple conditions must be true for the overall statement to be true.

Let's represent this using the variables P and Q, where P could stand for 'The weather is sunny' and Q for 'The park is free'. The logical conjunction of P and Q is denoted as P \( \wedge \) Q. In a truth table, this conjunction is only true when both P and Q are individually true. If either is false, the conjunction is false. This idea is fundamental in logic problems, like the one in our textbook exercise, and is symbolized by the 'and' gate in circuit design as well.

Ensuring students grasp this concept allows them to better understand more complex logical statements involving multiple variables and operators.

Logical Disjunction

Now let's say for your picnic, you're happy if you have either sandwiches or pizza. This 'or' condition is what we call a logical disjunction in propositional logic. It represents situations where at least one of multiple conditions must be true for the overall statement to be true.

Using our variables P and Q from earlier, the logical disjunction is written as P \( \vee \) Q. The truth table for a disjunction tells us that the statement P \( \vee \) Q is true if at least one of P or Q is true. The disjunction is only false if both P and Q are false. In our exercise, disjunction plays a crucial role in deconstructing the expression P \( \wedge \) (Q \( \vee \) R) into a form that can be compared with its equivalent form.

Propositional Logic

The backbone of our problem-solving exercise is propositional logic, a branch of logic that deals with propositions which can either be true or false. These propositions are represented by symbols like P, Q, and R in our exercise.

The beauty of propositional logic lies in its ability to construct complex expressions using simple logical operations like conjunction (\( \wedge \) - logical 'and') and disjunction (\( \vee \) - logical 'or'). By understanding the basic operations, students can build and deconstruct logical statements to analyze and solve intricate logical puzzles.

Improving students' understanding of propositional logic enhances their ability to approach logical reasoning in mathematics, computer science, and even daily decision-making processes.

Logical Equivalence

At the crux of our textbook exercise is proving that two logical expressions represent the same truth under all circumstances, known as logical equivalence. We say two propositions are logically equivalent if they always yield the same truth value in a truth table.

In our situation, we're showing the equivalence of P \( \wedge \) (Q \( \vee \) R) and (P \( \wedge \) Q) \( \vee \) (P \( \wedge \) R). To prove this, we examine if both expressions have identical truth values for all possible true and false combinations of P, Q, and R. If they match perfectly across the table, we've demonstrated logical equivalence.

This concept is not only pivotal in abstract logic but also in areas like digital circuit design and software development, where ensuring functional equivalence between different implementations of the same logic can be crucial.