p: 10+6=15 Determine the truth value for the statement pv-p. Choose the correct truth value below. Opv-p is false. Opv-p is true.
p: 10+6=15 Determine the truth value for the statement pv-p. Choose the correct truth value below. Opv-p is false. Opv-p is true.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Logical Reasoning and Truth Values**
**Statement Definition:**
Let \( p \) represent the following statement.
\[ p: 10 + 6 = 15 \]
**Task:**
Determine the truth value for the statement \( p \vee \sim p \).
**Options:**
- \( p \vee \sim p \) is false.
- \( p \vee \sim p \) is true.
**Explanation:**
In logical reasoning, \( p \) represents a proposition which can be either true or false. The notation \( \sim p \) denotes the negation of \( p \). The expression \( p \vee \sim p \) is a logical disjunction, meaning it is true if at least one of the components (\( p \) or \( \sim p \)) is true.
Since a proposition and its negation together encompass all possibilities (either \( p \) is true or \( \sim p \) is true), the disjunction \( p \vee \sim p \) is always true, known as the law of excluded middle. Here, the specific statement \( p: 10 + 6 = 15 \) is true, but regardless of its truth value on its own, the expression \( p \vee \sim p \) is universally true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7084a03c-72b7-406e-bd8b-9d3746e428bd%2F20af0d65-1e06-4c3c-b5f5-d7ba7b4b3868%2F0m7facr_processed.png&w=3840&q=75)
Transcribed Image Text:**Logical Reasoning and Truth Values**
**Statement Definition:**
Let \( p \) represent the following statement.
\[ p: 10 + 6 = 15 \]
**Task:**
Determine the truth value for the statement \( p \vee \sim p \).
**Options:**
- \( p \vee \sim p \) is false.
- \( p \vee \sim p \) is true.
**Explanation:**
In logical reasoning, \( p \) represents a proposition which can be either true or false. The notation \( \sim p \) denotes the negation of \( p \). The expression \( p \vee \sim p \) is a logical disjunction, meaning it is true if at least one of the components (\( p \) or \( \sim p \)) is true.
Since a proposition and its negation together encompass all possibilities (either \( p \) is true or \( \sim p \) is true), the disjunction \( p \vee \sim p \) is always true, known as the law of excluded middle. Here, the specific statement \( p: 10 + 6 = 15 \) is true, but regardless of its truth value on its own, the expression \( p \vee \sim p \) is universally true.
Expert Solution
Step 1
Remember the truth values of the following :
| True | False |
| False | True |
and
| True | True | True |
| True | False | True |
| False | True | True |
| False | False | False |
Step by step
Solved in 3 steps
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