Problem 10 - Thinking Mathematically

Question

Use a truth table to determine whether the two statements are equivalent. \((p \vee q) \wedge r, p \vee(q \wedge r)\)

Step-by-Step Solution

Answer

By following the mentioned process and evaluating each possible combination of truth values for \(p, q, r\), one can determine whether the two expressions \((p \vee q) \wedge r\) and \(p \vee(q \wedge r)\) are equivalent.

Step 1: Define the variables

Begin by defining the three variables \(p, q, r\). Each of these variables can hold either a true (T) or false (F) value.

Step 2: Create the truth table

Create a truth table that includes all possible combinations of true and false values for \(p, q, r\). This will result in \(2^3 = 8\) different cases.

Step 3: Calculate the truth values of the first statement

For each row in the truth table, calculate the truth value of the first expression \((p \vee q) \wedge r\). The operation \(\vee\) stands for 'or' such that \(p \vee q\) is true if either \(p\) or \(q\) is true. The operation \(\wedge\) stands for 'and' such that \((p \vee q) \wedge r\) is true if both \(p \vee q\) and \(r\) are true.

Step 4: Calculate the truth values of the second statement

For each row in the truth table, calculate the truth value of the second expression \(p \vee(q \wedge r)\). Here, \(q \wedge r\) is true if both \(q\) and \(r\) are true. Then, \(p \vee(q \wedge r)\) is true if either \(p\) or \((q \wedge r)\) is true.

Step 5: Compare the results

Compare the truth values of the first and second expressions for each row in the truth table. If the truth values are the same for each combination of \(p, q, r\), then the expressions are equivalent. If not, they are not equivalent.

Key Concepts

Understanding Truth Tables

Truth tables are a systematic way to explore how different logical expressions are evaluated. They help us see the outcomes of expressions based on different inputs. In our problem, we use a truth table to compare two logical statements:

• \((p \vee q) \wedge r\)
• \(p \vee(q \wedge r)\)

To create a truth table, list all possible combinations of truth values for the given variables. For three variables \(p, q, r\), you'll have \(2^3 = 8\) combinations. For each row, compute the truth value of both expressions. By comparing each row's results, you determine if the statements are logically equivalent.

Exploring Logical Operators

Logical operators are symbols or words used to connect statements in logical expressions. In our exercise:

• \(\vee\) represents 'or'. It's true if at least one of the connected statements is true.
• \(\wedge\) stands for 'and'. It's true only if both connected statements are true.

Understanding these operators is crucial for evaluating each part of a logical expression. For instance, in \((p \vee q)\), it's enough for either \(p\) or \(q\) to be true. But in \((p \vee q) \wedge r\), both \((p \vee q)\) and \(r\) need to be true for the whole expression to be true.

Diving into Propositional Logic

Propositional logic is the branch of logic dealing with propositions that can be true or false. In this exercise, expressions like \((p \vee q) \wedge r\) and \(p \vee(q \wedge r)\) are propositional formulas.
A key goal in propositional logic is determining logical equivalence. This involves checking if different expressions always lead to the same truth values under all interpretations of their components. By using truth tables, we systematically compare these values to see if two logical formulas are, in fact, logically equivalent.

Utilizing Boolean Algebra

Boolean algebra is a mathematical way to work with logical values using operators like those in our exercise. Originating from the work of George Boole, it deals with binary variables and is foundational for computer science.
In Boolean algebra, expressions are simplified using laws and properties, like the distributive, associative, and commutative laws. Calculating whether \((p \vee q) \wedge r\) is equivalent to \(p \vee(q \wedge r)\) can also be approached through Boolean algebra methods. By structuring our understanding with Boolean laws, we often find simpler ways to express or validate logical statements beyond traditional truth tables.