For Problems 25-40, use the indirect approach to test each argument for validity. If the argument is invalid, produce a counterexample chart. 25.Iv p 26. (~w b)^(~wvs) (dv m)→p lv-d b. 27. (cvm)→s 28. p^(qvr) (-ca)→m (av w)→~q S. 29. p k 30. ad -p(WAS) -dv p -Sv p P(CAb) b-(a-c) k
For Problems 25-40, use the indirect approach to test each argument for validity. If the argument is invalid, produce a counterexample chart. 25.Iv p 26. (~w b)^(~wvs) (dv m)→p lv-d b. 27. (cvm)→s 28. p^(qvr) (-ca)→m (av w)→~q S. 29. p k 30. ad -p(WAS) -dv p -Sv p P(CAb) b-(a-c) k
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
100%
Solve 27 to 33 odd indirect method
![**Page 128 - Arguments**
### Problems
**For Problems 15–18, answer true or false.**
15. When using the indirect approach to test for validity, if there are no contradictions, the argument is valid.
16. When using the indirect approach to test for validity, if the conclusion is a conditional statement, we assume the LHS and the RHS are both false.
17. When using the indirect approach to test for validity, if the conclusion is a disjunction, we assume both of its components are false.
18. When using the indirect approach to test for validity, if we find a contradiction, we then produce a counterexample chart.
---
**Problems 19–24 each represent the conclusion to an argument. If we use the indirect approach to test each argument for validity, what would be the truth value of m in each case?**
19. \( a \lor \sim m \)
20. \( m \rightarrow p \)
21. \( (a \land b) \rightarrow (m \lor k) \)
22. \( \sim (m \rightarrow c) \rightarrow j \)
23. \( \sim (m \land q) \lor (m \land x) \)
24. \( (m \rightarrow s) \rightarrow (\sim s \lor h) \)
---
**For Problems 25–40, use the indirect approach to test each argument for validity. If the argument is invalid, produce a counterexample chart.**
25.
\[
\begin{align*}
l \lor \sim p \\
(d \lor m) \rightarrow p \\
l \lor \sim d
\end{align*}
\]
26.
\[
\begin{align*}
(\sim w \rightarrow b) \land (\sim w \lor s) \\
b \leftrightarrow s \\
\underline{b}
\end{align*}
\]
27.
\[
\begin{align*}
(c \lor m) \rightarrow s \\
(\sim c \rightarrow a) \rightarrow m \\
\underline{s}
\end{align*}
\]
28.
\[
\begin{align*}
p \land (q \lor r) \\
(a \lor w) \rightarrow \sim q \\
\underline{r}
\end{align*}
\]
29.
\[
\begin{align](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50969b9e-0557-47b3-9c06-ed63c28b991b%2Fe7252cf2-ca92-4a78-8abc-5ccff2eff150%2F32ug4hd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Page 128 - Arguments**
### Problems
**For Problems 15–18, answer true or false.**
15. When using the indirect approach to test for validity, if there are no contradictions, the argument is valid.
16. When using the indirect approach to test for validity, if the conclusion is a conditional statement, we assume the LHS and the RHS are both false.
17. When using the indirect approach to test for validity, if the conclusion is a disjunction, we assume both of its components are false.
18. When using the indirect approach to test for validity, if we find a contradiction, we then produce a counterexample chart.
---
**Problems 19–24 each represent the conclusion to an argument. If we use the indirect approach to test each argument for validity, what would be the truth value of m in each case?**
19. \( a \lor \sim m \)
20. \( m \rightarrow p \)
21. \( (a \land b) \rightarrow (m \lor k) \)
22. \( \sim (m \rightarrow c) \rightarrow j \)
23. \( \sim (m \land q) \lor (m \land x) \)
24. \( (m \rightarrow s) \rightarrow (\sim s \lor h) \)
---
**For Problems 25–40, use the indirect approach to test each argument for validity. If the argument is invalid, produce a counterexample chart.**
25.
\[
\begin{align*}
l \lor \sim p \\
(d \lor m) \rightarrow p \\
l \lor \sim d
\end{align*}
\]
26.
\[
\begin{align*}
(\sim w \rightarrow b) \land (\sim w \lor s) \\
b \leftrightarrow s \\
\underline{b}
\end{align*}
\]
27.
\[
\begin{align*}
(c \lor m) \rightarrow s \\
(\sim c \rightarrow a) \rightarrow m \\
\underline{s}
\end{align*}
\]
28.
\[
\begin{align*}
p \land (q \lor r) \\
(a \lor w) \rightarrow \sim q \\
\underline{r}
\end{align*}
\]
29.
\[
\begin{align
Transcribed Image Text:# In-Class Exercises and Problems for Section 2.5
## Exercise Problems:
### Problem 31
1. \( \sim p \rightarrow (d \rightarrow \sim z) \)
2. \( p \rightarrow q \)
3. \( c \vee d \)
4. \( \sim q \)
Conclusion: \( z \vee c \)
### Problem 32
1. \( \sim p \rightarrow \sim n \)
2. \( r \rightarrow \sim s \)
3. \( n \)
4. \( \sim p \vee (\sim s \rightarrow \sim q) \)
Conclusion: \( q \vee r \)
### Problem 33
1. \( (s \vee w) \rightarrow b \)
2. \( (b \wedge k) \leftrightarrow (a \rightarrow r) \)
3. \( a \vee p \)
Conclusion: \( (s \wedge k) \rightarrow (r \vee p) \)
### Problem 34
1. \( a \rightarrow (r \vee b) \)
2. \( \sim b \vee g \)
3. \( (r \vee g) \leftrightarrow c \)
Conclusion: \( a \rightarrow c \)
### Problem 35
1. \( p \vee \sim c \)
2. \( (j \vee b) \rightarrow a \)
3. \( (b \wedge p) \vee k \)
4. \( (k \vee c) \rightarrow j \)
Conclusion: \( p \leftrightarrow a \)
### Problem 36
1. \( h \rightarrow (a \vee p) \)
2. \( s \rightarrow (\sim p \vee c) \)
3. \( (a \vee s) \rightarrow d \)
Conclusion: \( b \rightarrow (c \vee d) \)
### Problem 37
1. \( (n \vee p) \rightarrow s \)
2. \( q \rightarrow (s \wedge w) \)
3. \( (m \wedge a) \leftrightarrow (\sim c \rightarrow n) \)
4. \( (q \vee \sim c) \rightarrow a \)
Conclusion: \( m \vee (p \rightarrow \sim w) \)
### Problem 38
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