Problem 8 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The two statements \((p \vee q) \vee r\) and \(p \vee (q \vee r)\) are equivalent, as their truth tables are the same.

Step 1 Define Variables

First, define the Boolean variables \(p\), \(q\), and \(r\). Each can either be true (T) or false (F), giving 2^3 = 8 possible combinations.

Step 2 Define the Operations

Define the two operations: \(A = (p \vee q) \vee r\) and \(B = p \vee (q \vee r)\). Notice that \( \vee \) stands for logical OR operator, which returns true if at least one of the operands is true.

Step 3 Construct and Fill the Truth Table

Construct a truth table and fill it out. You should end up with a table of the combinations of truth values for p, q, r, A, and B.

Step 4 Compare the Results

Compare the results of A and B for each row of the truth table. If they are the same for each possible combination of truth values for p, q, r, then the two statements are equivalent.

Key Concepts

truth table

A truth table is a mathematical table used to determine the output of logical expressions based on their inputs. In logic, truth tables help us understand the outcomes of combinations of logical operations. To create a truth table, you need to list all possible combinations of truth values for the variables involved.

For instance, if you have three Boolean variables: \(p\), \(q\), and \(r\), each can be true (\(T\)) or false (\(F\)). This means there are \(2^3 = 8\) possible combinations of truth values. The truth table will include columns for each of these variables and additional columns for the expressions being evaluated.

• Column 1: Possible values for \(p\)
• Column 2: Possible values for \(q\)
• Column 3: Possible values for \(r\)
• Column 4: Result for expression \((p \vee q) \vee r\)
• Column 5: Result for expression \(p \vee (q \vee r)\)

By reviewing the results under these conditions, you can determine whether different logical statements are equivalent. In our example, you would compare expressions to confirm if they yield the same output across all combinations.

logical OR operator

The logical OR operator, symbolized by \(\vee\), is a fundamental concept in Boolean algebra. It is used in logical expressions to combine two or more Boolean variables or values. The logical OR operation results in true when any one of its operands is true.

Here's how it works for two operands:\(A\) and \(B\):

• If both \(A\) and \(B\) are false (\(F\)), the result is false (\(F\)).
• If either \(A\) or \(B\) is true (\(T\)), or if both are true, the result is true (\(T\)).

For example, in a truth table for the operation \(p \vee q\), the result is true when at least one of \(p\) or \(q\) is true. This principle applies to more complex expressions such as \((p \vee q) \vee r\). In this case, the expression evaluates to true if \(p\), \(q\), or \(r\) is true.

Boolean variables

Boolean variables are the basic units of logic used in operations involving truth values. They can hold one of two possible values: true or false. These values are often represented as \(T\) and \(F\), respectively. Boolean variables make up the building blocks for more complex logical expressions and operations such as AND, OR, and NOT.

In logical expressions, these variables are manipulated using logical operators to derive conclusions or check the equivalence of statements. For example, when defining expressions like \((p \vee q) \vee r\) and \(p \vee (q \vee r)\), each of \(p\), \(q\), and \(r\) are Boolean variables.

• \(p = T\) indicates that the statement represented by \(p\) is true.
• \(p = F\) indicates that the statement represented by \(p\) is false.

Understanding Boolean variables is essential in constructing truth tables and logical expressions, allowing for clear representation and manipulation of logical statements.