https://r2.erweima.ai/mathbot/image/17269059288068rol35wu.jpg

Let's break down the problem step by step.

We are given that:

1. \( p \land q = 1 \), which means both \( p \) and \( q \) are true (since \( p \land q \) is the logical "and", both conditions must hold true for the result to be 1).
   
Now, we need to evaluate the truth values of the following compound propositions to determine which ones are true (i.e., have a truth value of 1):

- I. \( \neg p \land q \)
- II. \( \neg p \lor q \)
- III. \( p \iff \neg q \)

### Analyzing each statement:

#### Proposition I: \( \neg p \land q \)
- \( p \land q = 1 \) implies that \( p = 1 \) and \( q = 1 \).
- \( \neg p \) is the negation of \( p \), so \( \neg p = 0 \).
- \( \neg p \land q = 0 \land 1 = 0 \).

Thus, Proposition I is **false**.

#### Proposition II: \( \neg p \lor q \)
- We already have \( \neg p = 0 \) and \( q = 1 \).
- \( \neg p \lor q = 0 \lor 1 = 1 \).

Thus, Proposition II is **true**.

#### Proposition III: \( p \iff \neg q \)
- \( p = 1 \) and \( \neg q = 0 \).
- \( p \iff \neg q \) is a biconditional statement, meaning it is true if both sides are equal. Here, \( 1 \iff 0 = 0 \).

Thus, Proposition III is **false**.

### Conclusion:
The only true proposition is **II**. Therefore, the correct answer is **B** (Yalnız II).

---

Would you like a more detailed explanation on any part of this problem or related topics? 

Here are 5 follow-up questions for you:
1. What does the logical operator \( \land \) represent in compound statements?
2. How do negations affect truth values in logical expressions?
3. Can you explain the biconditional (\( \iff \)) operator with examples?
4. What are the truth tables for logical operations like \( \land \), \( \lor \), and \( \iff \)?
5. How would the result change if \( p \land q = 0 \)?

**Tip**: In logic, understanding how each logical operator affects truth values is essential. Keep practicing truth tables to master compound propositions!

Given that p ∧ q = 1, determine which of the following propositions have a truth value of 1:
I. ¬p ∧ q
II. ¬p ∨ q
III. p ⇔ ¬q

This problem involves analyzing compound logical statements based on the given condition p ∧ q = 1. Explore how different logical operators, such as negation (¬), conjunction (∧), disjunction (∨), and biconditional (⇔), affect the truth values of propositions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation