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Magazine Chapter 1: Problem 13
Verify each, where \(f\) denotes a contradiction. $$(p \vee q) \rightarrow r \equiv(p \rightarrow r) \wedge(q \rightarrow r)$$
Short Answer
Expert verified
Using the truth table method, we found that the columns for \((p \vee q) \rightarrow r\) and \((p \rightarrow r) \wedge (q \rightarrow r)\) are identical, thus proving the given equivalence is true:
$$(p \vee q) \rightarrow r \equiv (p \rightarrow r) \wedge (q \rightarrow r)$$
Step by step solution
01
Set up the truth table template
First, we will set up a template for the truth table with columns for \(p\), \(q\), and \(r\), as well as columns for the left-side expression and right-side expression of the equivalence.
| \(p\) | \(q\) | \(r\) | \((p \vee q) \rightarrow r\) | \((p \rightarrow r) \wedge (q \rightarrow r)\) |
|-----|-----|-----|-------------------------|---------------------------------------------|
02
Fill in the values for \(p\), \(q\), and \(r\)
In this step, we will fill in all possible combinations of truth values for \(p\), \(q\), and \(r\):
| \(p\) | \(q\) | \(r\) | \((p \vee q) \rightarrow r\) | \((p \rightarrow r) \wedge (q \rightarrow r)\) |
|-----|-----|-----|-------------------------|---------------------------------------------|
| T | T | T | | |
| T | T | F | | |
| T | F | T | | |
| T | F | F | | |
| F | T | T | | |
| F | T | F | | |
| F | F | T | | |
| F | F | F | | |
03
Calculate the values for the left-side expression
Now, we will calculate the truth values for the left-side expression in the equivalence, \((p \vee q) \rightarrow r\):
| \(p\) | \(q\) | \(r\) | \((p \vee q) \rightarrow r\) | \((p \rightarrow r) \wedge (q \rightarrow r)\) |
|-----|-----|-----|-------------------------|---------------------------------------------|
| T | T | T | T | |
| T | T | F | F | |
| T | F | T | T | |
| T | F | F | F | |
| F | T | T | T | |
| F | T | F | F | |
| F | F | T | T | |
| F | F | F | T | |
04
Calculate the values for the right-side expression
Next, we will calculate the truth values for the right-side expression in the equivalence, \((p \rightarrow r) \wedge (q \rightarrow r)\):
| \(p\) | \(q\) | \(r\) | \((p \vee q) \rightarrow r\) | \((p \rightarrow r) \wedge (q \rightarrow r)\) |
|-----|-----|-----|-------------------------|---------------------------------------------|
| T | T | T | T | T |
| T | T | F | F | F |
| T | F | T | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | T | F | F | F |
| F | F | T | T | T |
| F | F | F | T | T |
05
Compare the columns for the left-side and right-side expressions
Finally, we will compare the columns for the left-side and the right-side expressions. If these columns are identical, then the equivalence is true.
| \(p\) | \(q\) | \(r\) | \((p \vee q) \rightarrow r\) | \((p \rightarrow r) \wedge (q \rightarrow r)\) |
|-----|-----|-----|-------------------------|---------------------------------------------|
| T | T | T | T | T |
| T | T | F | F | F |
| T | F | T | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | T | F | F | F |
| F | F | T | T | T |
| F | F | F | T | T |
Since the columns for \((p \vee q) \rightarrow r\) and \((p \rightarrow r) \wedge (q \rightarrow r)\) are identical, the given equivalence is true:
$$(p \vee q) \rightarrow r \equiv (p \rightarrow r) \wedge (q \rightarrow r)$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logical Equivalence
Logical equivalence is a fundamental concept in propositional logic. It is significant because it allows us to transform propositions into equivalent forms, which can simplify logical reasoning and proofs. Two statements are logically equivalent if they have the same truth value in every possible scenario.
When dealing with logical equivalence, it's essential to understand that it doesn't matter what symbols or words we use to express these propositions, as long as they yield the same truth value under all circumstances. This is what makes logical equivalence particularly useful in areas like computer science, mathematics, and philosophy.
In this specific exercise, the goal is to prove that \[(p \vee q) \rightarrow r\] is logically equivalent to\[(p \rightarrow r) \wedge (q \rightarrow r)\].
By using truth tables, we can systematically show and verify this equivalence.
When dealing with logical equivalence, it's essential to understand that it doesn't matter what symbols or words we use to express these propositions, as long as they yield the same truth value under all circumstances. This is what makes logical equivalence particularly useful in areas like computer science, mathematics, and philosophy.
In this specific exercise, the goal is to prove that \[(p \vee q) \rightarrow r\] is logically equivalent to\[(p \rightarrow r) \wedge (q \rightarrow r)\].
By using truth tables, we can systematically show and verify this equivalence.
Truth Tables
Truth tables are a handy tool in propositional logic for checking the truth value of complex expressions. By laying out all possible truth values of the basic variables and evaluating the expressions, we can verify logical equivalences and other properties.
To construct a truth table, follow these steps:
To construct a truth table, follow these steps:
- List all variables involved in the expressions. Each variable can be either true (T) or false (F).
- Construct a table where each row represents one possible combination of truth values.
- Add columns for each logical expression you need to evaluate.
- Fill these columns based on the truth values of the variables in each row.
Logical Connectives
Logical connectives are the building blocks of logical expressions. They allow us to connect simple statements or propositions into more complex forms. Knowing how these connectives work enables us to construct and deconstruct logical expressions.
Key logical connectives used in this exercise include:
Key logical connectives used in this exercise include:
- Conjunction (\(\land\)): True only when both propositions are true.
- Disjunction (\(\lor\)): True when at least one of the propositions is true.
- Conditional (\(\rightarrow\)): True unless a true proposition leads to a false one.
Conditional Statements
Conditional statements are unique and crucial types of logical expressions occurring frequently in propositional logic. They follow the 'if-then' format, represented as \(p \rightarrow q\), where \(p\) is a hypothesis and \(q\) is a conclusion.
The truth of a conditional statement depends on the relationship between the hypothesis and the conclusion:
The truth of a conditional statement depends on the relationship between the hypothesis and the conclusion:
- The statement is true if both the hypothesis and conclusion are true.
- It is also true if the hypothesis is false, regardless of the conclusion's truth value.
- It is only false when a true hypothesis leads to a false conclusion.
