Problem 10
Question
Construct a truth table for the given statement. \(p \rightarrow(q \vee r)\)
Step-by-Step Solution
Verified Answer
The truth table for the logical statement 'p implies (q or r)' will consist of five columns representing 'p', 'q', 'r', '(q ∨ r)', and '\(p \rightarrow (q \vee r)\)'. There will be eight rows to account for all the possible combinations of truth values for 'p', 'q', and 'r'. The truth values of '(q ∨ r)' and '\(p \rightarrow (q \vee r)\)' are then calculated based on the truth values of 'p', 'q', and 'r' and the rules of logical implication and 'or'.
1Step 1: Define the Variables
Set up a truth table with four columns, one for each variable (p, q, r) and one for the compound statement. Each variable can take on the value of true (T) or false (F), so there should be 8 different rows to account for all possible combinations of true and false for the three variables.
2Step 2: Calculate q ∨ r
Once the truth values for p, q, and r are listed in the first three columns of the table, the next step is to calculate the value of 'q or r' using the 'or' operation rules. In this operation, the result is true if at least one of 'q' or 'r' is true; it is false otherwise. Put the results in the fourth column.
3Step 3: Calculate p → (q ∨ r)
Repeat the process using the 'implies' rule (→). According to the 'if then' operation rules, a formula is false only if the first part is true and the second is false. It is true in all other conditions. Compute the truth values of \(p \rightarrow (q \vee r)\) using the values of 'p' from the first column and the calculated values of '(q ∨ r)' immediately obtained in step 2. The result should be placed in the fifth column.
Key Concepts
Logical OperationsTruth ValuesCompound Statements
Logical Operations
Logical operations are foundational tools in constructing truth tables. These operations manipulate truth values, typically represented as True (T) or False (F), in specific ways depending on the type of logical operation used. In the given exercise, we encounter two key logical operations: the logical 'OR' (denoted as \(\vee\)) and the logical 'IMPLIES' (denoted as \(\rightarrow\)).
- The logical 'OR' operation results in True if at least one of the operands is True. It will only result in False if both operands are False.
- The logical 'IMPLIES' operation results in False only if the first operand is True and the second is False. Otherwise, it remains True.
Truth Values
Truth values are simply the outcomes of logical operations and are essential in evaluating logical statements. These values, True (T) or False (F), correspond to the real-life conditions or propositions being examined. In building a truth table, different combinations of these truth values are the basis for the logical evaluations.
In our example, you start by listing all possible combinations of truth values for each variable involved in the statement \(p \rightarrow(q \vee r)\). You carefully calculate the resulting values for each operation step by step, which ultimately helps in verifying the truthfulness of the entire compound statement.
While individual variables (like \(p\), \(q\), \(r\)) hold these truth values, compound statements will dependably reflect a truth value based on these foundational evaluations.
In our example, you start by listing all possible combinations of truth values for each variable involved in the statement \(p \rightarrow(q \vee r)\). You carefully calculate the resulting values for each operation step by step, which ultimately helps in verifying the truthfulness of the entire compound statement.
While individual variables (like \(p\), \(q\), \(r\)) hold these truth values, compound statements will dependably reflect a truth value based on these foundational evaluations.
Compound Statements
Compound statements are created when multiple simple statements are combined using logical operations. They allow for more complex reasoning and expand our ability to evaluate comprehensive logical scenarios.
In the exercise, the compound statement \(p \rightarrow(q \vee r)\) involves two operations. Initially, the inner expression \(q \vee r\) is evaluated, joining \(q\) and \(r\) with an 'OR'. Then, the result is combined with \(p\) using an 'IMPLIES'.
Creating a truth table for such statements involves multiple steps, ensuring each component of the compound statement is accurately calculated. Ultimately, this process allows us to fully understand the logical proposition, and correctly determine its overall truth value for every possible scenario of truth values for the individual variables.
In the exercise, the compound statement \(p \rightarrow(q \vee r)\) involves two operations. Initially, the inner expression \(q \vee r\) is evaluated, joining \(q\) and \(r\) with an 'OR'. Then, the result is combined with \(p\) using an 'IMPLIES'.
Creating a truth table for such statements involves multiple steps, ensuring each component of the compound statement is accurately calculated. Ultimately, this process allows us to fully understand the logical proposition, and correctly determine its overall truth value for every possible scenario of truth values for the individual variables.
Other exercises in this chapter
Problem 10
Write the negation of each conditional statement. \(\sim p \rightarrow r\)
View solution Problem 10
Use a truth table to determine whether the two statements are equivalent. \((p \vee q) \wedge r, p \vee(q \wedge r)\)
View solution Problem 10
Let \(p\) and q represent the following statements: $$ \begin{aligned} &p: 4+6=10 \\ &q: 5 \times 8=80 \end{aligned} $$ Determine the truth value for each state
View solution Problem 10
Let \(p\) and q represent the following simple statements: p: I study. \(q:\) I pass the course. Write each compound statement in symbolic form. I do not study
View solution