What conclusion can be drawn from the following? State the property used (transitive, symmetry, contrapositive, etc.). ~c→~ f, g⇒b, p→f, c→~b
What conclusion can be drawn from the following? State the property used (transitive, symmetry, contrapositive, etc.). ~c→~ f, g⇒b, p→f, c→~b
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 10CT: Statement P and Q are true while R is a false statement. Classify as true or false:...
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![### Logical Reasoning: Understanding Properties in Propositional Logic
**Question:**
What conclusion can be drawn from the following? State the property used (transitive, symmetry, contrapositive, etc.).
\[
\sim c \rightarrow \sim f, \quad g \rightarrow \, b, \quad p \rightarrow f, \quad \text{c} \rightarrow\, b
\]
**Explanation:**
The problem involves understanding relationships and properties of propositional logic statements. Each statement given implies a certain logical relationship. The statements can be interpreted as follows:
1. \(\sim c \rightarrow \sim f\): If \(c\) is false, then \(f\) is false.
2. \(g \rightarrow \, b\): If \(g\) is true, then \(b\) is true.
3. \(p \rightarrow f\): If \(p\) is true, then \(f\) is true.
4. \(c \rightarrow\, b\): If \(c\) is true, then \(b\) is true.
**Analyzing Properties and Drawing Conclusions:**
1. **Contrapositive Property:**
- For the statement \(c \rightarrow b\), its contrapositive is:
\[\sim b \rightarrow \sim c\]
- This means that if \(b\) is false, then \(c\) is also false.
2. **Combining Statements:**
- Given \(c \rightarrow b\) and the contrapositive \(\sim b \rightarrow \sim c\), we establish a direct relationship between \(c\) and \(b\).
- Next, using \(\sim c \rightarrow \sim f\), which says if \(c\) is false, then \(f\) is false, we infer using contrapositive logic:
\[\sim f \rightarrow \sim \sim c\]
which simplifies to:
\[\sim f \rightarrow c\]
3. **Transitive Property:**
- Since we have \(c \rightarrow b\) and from the contrapositive property \(\sim f \rightarrow c\), with the additional statement \(g \rightarrow b\), we can further determine if there exists mutual relationships or implied conclusions among these propositions.
Given these deductions, the overall examination revolves around integrating various logical properties:
- Contrapositive Property: To find](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d94fb0e-3cb9-42c0-8e4a-cf5f8f1ded5d%2Ffa1392a8-81d7-47ef-8fe4-838776e8669e%2F27qgyi_processed.png&w=3840&q=75)
Transcribed Image Text:### Logical Reasoning: Understanding Properties in Propositional Logic
**Question:**
What conclusion can be drawn from the following? State the property used (transitive, symmetry, contrapositive, etc.).
\[
\sim c \rightarrow \sim f, \quad g \rightarrow \, b, \quad p \rightarrow f, \quad \text{c} \rightarrow\, b
\]
**Explanation:**
The problem involves understanding relationships and properties of propositional logic statements. Each statement given implies a certain logical relationship. The statements can be interpreted as follows:
1. \(\sim c \rightarrow \sim f\): If \(c\) is false, then \(f\) is false.
2. \(g \rightarrow \, b\): If \(g\) is true, then \(b\) is true.
3. \(p \rightarrow f\): If \(p\) is true, then \(f\) is true.
4. \(c \rightarrow\, b\): If \(c\) is true, then \(b\) is true.
**Analyzing Properties and Drawing Conclusions:**
1. **Contrapositive Property:**
- For the statement \(c \rightarrow b\), its contrapositive is:
\[\sim b \rightarrow \sim c\]
- This means that if \(b\) is false, then \(c\) is also false.
2. **Combining Statements:**
- Given \(c \rightarrow b\) and the contrapositive \(\sim b \rightarrow \sim c\), we establish a direct relationship between \(c\) and \(b\).
- Next, using \(\sim c \rightarrow \sim f\), which says if \(c\) is false, then \(f\) is false, we infer using contrapositive logic:
\[\sim f \rightarrow \sim \sim c\]
which simplifies to:
\[\sim f \rightarrow c\]
3. **Transitive Property:**
- Since we have \(c \rightarrow b\) and from the contrapositive property \(\sim f \rightarrow c\), with the additional statement \(g \rightarrow b\), we can further determine if there exists mutual relationships or implied conclusions among these propositions.
Given these deductions, the overall examination revolves around integrating various logical properties:
- Contrapositive Property: To find
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