Problem 67 - Discrete Mathematics with Applications

Question

Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) \(0.5 .\) Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) $$p \vee r$$

Step-by-Step Solution

Answer

The truth value of the compound proposition \(p \vee r\) is 1, meaning it is true, given that \(t(p) = 1\) and \(t(r) = 0.5\).

Step 1: Recall the truth table for logical OR

The truth table for logical OR (\(\vee\)) is as follows: | A | B | A \(\vee\) B | |---|---|----------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 1 | The compound proposition is true if at least one of its component propositions is true.

Step 2: Determine the truth values of the component propositions

We know that \(t(p) = 1\) and \(t(r) = 0.5\). Since \(t(p)\) is 1, \(p\) is true. Since \(t(r)\) is greater than 0, \(r\) is also considered true.

Step 3: Apply the truth table of logical OR to the given truth values

The truth values of the component propositions are as follows: | p | r | |---|---| | 1 | 1 | Using the truth table of logical OR, we find the truth value of the compound proposition \(p \vee r\): | A | B | A \(\vee\) B | |---|---|----------| | 1 | 1 | 1 |

Step 4: Determine the truth value of \(p \vee r\)

Based on the truth table for logical OR, the truth value of the compound proposition \(p \vee r\) is 1, which means it is true.

Key Concepts

Truth Tables

Truth tables are a fundamental tool in propositional logic. They help us understand how different logical operations work by showing all possible true or false combinations of the involved propositions.
Consider the proposition "A or B" using the logical OR operator, denoted by the symbol \( \vee \). The truth table for this operation would present every possible combination of truth values for "A" and "B", and the result of the operation.
Here's why truth tables are so useful:

• They provide a clear and systematic way to calculate the truth value of logical expressions.
• Utilize them to determine the outcome of compound propositions.
• They are essential for verifying the equivalence of logical statements.

For example, if "A" is true (1) and "B" is false (0), the table shows that "A or B" is true. Truth tables make such evaluations simple and visual.

Logical OR

The logical OR operation is a cornerstone of propositional logic. It takes two propositions as inputs and returns true if at least one of the propositions is true. The logical OR operator in propositional logic is represented by \( \vee \). Here's a quick breakdown:

• \(A \vee B\) is true if either "A" or "B" is true, or if both are true.
• The only time \(A \vee B\) is false is when both "A" and "B" are false.

To see logical OR in action, imagine evaluating the statement \(p \vee r\) with \(t(p) = 1\) and \(t(r) = 0.5\). Both propositions have non-zero truth values, meaning they are considered true. Thus, \(p \vee r\) evaluates to true as well. Logical OR simplifies complex logical expressions by combining them in a way that captures the inclusive nature of logical statements.

Truth Value

In propositional logic, the truth value indicates whether a statement is true or false. It is commonly represented as 1 (true) or 0 (false). However, other values can indicate degrees of truth, as in our exercise where \(t(q) = 0.3\) and \(t(r) = 0.5\).
Truth values are the backbone of logic as they help:

• Determine the overall truth of complex propositions.
• Construct and evaluate logical arguments.
• Illustrate how propositions interact through different logical operators.

For example, when evaluating \(p \vee r\), you were given the truth values \(t(p) = 1\) (true) and \(t(r) = 0.5\). Both propositions contributed positively to the compound expression, resulting in a truth value of 1, or true, for \(p \vee r\). Understanding truth values is crucial for correctly applying logical operations and accurately interpreting the results.