### Problem 3: Determine if each form is a tautology, a contradiction, or a contingency#### a) \[(p \land \neg q) \land (\neg p \lor q)\]Here, you need to determine whether the provided logical expression is always true (tautology), always false (contradiction), or sometimes true and sometimes false (contingency).#### b) \[(p \lor \neg q) \rightarrow (p \land q)\]Here, you need to determine whether the provided logical implication is always true (tautology), always false (contradiction), or sometimes true and sometimes false (contingency).#### c) \[(p \land \neg p) \rightarrow q\]Here, you need to determine whether the provided logical implication is always true (tautology), always false (contradiction), or sometimes true and sometimes false (contingency).---### Key Terms:- **Tautology**: An expression that is always true, regardless of the truth values of the variables involved.- **Contradiction**: An expression that is always false, regardless of the truth values of the variables involved.- **Contingency**: An expression that is neither a tautology nor a contradiction, meaning it is sometimes true and sometimes false depending on the truth values of the variables involved.To determine the nature of each logical form, you can construct a truth table or use logical equivalences to simplify the expressions. This analysis will help you to understand the type of logical statement each form represents.

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Answered: Determine if each form is a tautology,…

Math

Advanced Math

Determine if each form is a tautology, a contradiction, or a contingency:

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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Question

### Problem 3: Determine if each form is a tautology, a contradiction, or a contingency

#### a) \[(p \land \neg q) \land (\neg p \lor q)\]

Here, you need to determine whether the provided logical expression is always true (tautology), always false (contradiction), or sometimes true and sometimes false (contingency).

#### b) \[(p \lor \neg q) \rightarrow (p \land q)\]

Here, you need to determine whether the provided logical implication is always true (tautology), always false (contradiction), or sometimes true and sometimes false (contingency).

#### c) \[(p \land \neg p) \rightarrow q\]

Here, you need to determine whether the provided logical implication is always true (tautology), always false (contradiction), or sometimes true and sometimes false (contingency).

---

### Key Terms:
- **Tautology**: An expression that is always true, regardless of the truth values of the variables involved.
- **Contradiction**: An expression that is always false, regardless of the truth values of the variables involved.
- **Contingency**: An expression that is neither a tautology nor a contradiction, meaning it is sometimes true and sometimes false depending on the truth values of the variables involved.

To determine the nature of each logical form, you can construct a truth table or use logical equivalences to simplify the expressions. This analysis will help you to understand the type of logical statement each form represents.

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