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 # In-Class Exercises and Problems for Section 2.5## Exercise Problems:### Problem 311. \( \sim p \rightarrow (d \rightarrow \sim z) \)2. \( p \rightarrow q \)3. \( c \vee d \)4. \( \sim q \) Conclusion: \( z \vee c \)### Problem 321. \( \sim p \rightarrow \sim n \)2. \( r \rightarrow \sim s \)3. \( n \)4. \( \sim p \vee (\sim s \rightarrow \sim q) \) Conclusion: \( q \vee r \)### Problem 331. \( (s \vee w) \rightarrow b \)2. \( (b \wedge k) \leftrightarrow (a \rightarrow r) \)3. \( a \vee p \) Conclusion: \( (s \wedge k) \rightarrow (r \vee p) \)### Problem 341. \( a \rightarrow (r \vee b) \)2. \( \sim b \vee g \)3. \( (r \vee g) \leftrightarrow c \) Conclusion: \( a \rightarrow c \)### Problem 351. \( p \vee \sim c \)2. \( (j \vee b) \rightarrow a \)3. \( (b \wedge p) \vee k \)4. \( (k \vee c) \rightarrow j \) Conclusion: \( p \leftrightarrow a \)### Problem 361. \( h \rightarrow (a \vee p) \)2. \( s \rightarrow (\sim p \vee c) \)3. \( (a \vee s) \rightarrow d \) Conclusion: \( b \rightarrow (c \vee d) \)### Problem 371. \( (n \vee p) \rightarrow s \)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

Note: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question…

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

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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

# 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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