Under which conditions is the following compound statement "FALSE"?

﻿left parenthesis tilde p logical and r right parenthesis logical or left parenthesis q logical and tilde r right parenthesis﻿ 


p is true, q is true, r is true


p is false, q is true, r is true


p is true, q is true, r is false


p is false, q is false, r is true

To determine under which conditions the compound statement is "FALSE," let's analyze the logical expression:

\[
(\neg p \land r) \lor (q \land \neg r)
\]

Where:
- \( \neg p \) is the negation of \( p \) (true if \( p \) is false and vice versa).
- \( \land \) denotes "and."
- \( \lor \) denotes "or."
- \( \neg r \) is the negation of \( r \).

Let's evaluate the compound statement under the given conditions.

### 1. \( p \) is true, \( q \) is true, \( r \) is true
- \( \neg p = \) false
- \( \neg r = \) false

\[
(\text{false} \land \text{true}) \lor (\text{true} \land \text{false}) = \text{false} \lor \text{false} = \text{false}
\]

**This condition makes the statement FALSE.**

### 2. \( p \) is false, \( q \) is true, \( r \) is true
- \( \neg p = \) true
- \( \neg r = \) false

\[
(\text{true} \land \text{true}) \lor (\text{true} \land \text{false}) = \text{true} \lor \text{false} = \text{true}
\]

**This condition makes the statement TRUE.**

### 3. \( p \) is true, \( q \) is true, \( r \) is false
- \( \neg p = \) false
- \( \neg r = \) true

\[
(\text{false} \land \text{false}) \lor (\text{true} \land \text{true}) = \text{false} \lor \text{true} = \text{true}
\]

**This condition makes the statement TRUE.**

### 4. \( p \) is false, \( q \) is false, \( r \) is true
- \( \neg p = \) true
- \( \neg r = \) false

\[
(\text{true} \land \text{true}) \lor (\text{false} \land \text{false}) = \text{true} \lor \text{false} = \text{true}
\]

**This condition makes the statement TRUE.**

### Conclusion:
The compound statement is **FALSE** only under the first condition:
- \( p \) is true, \( q \) is true, \( r \) is true.

Would you like further details or have any questions?

Here are five related questions:
1. How would the truth value change if both \( p \) and \( q \) were false?
2. What is the truth value if \( p \) is false, \( q \) is false, and \( r \) is false?
3. How does the distributive property of logic apply to this expression?
4. What would the result be if the expression used \( \lor \) instead of \( \land \) inside the parentheses?
5. How would adding another logical term, like \( \neg q \), affect the statement?

**Tip:** When evaluating complex logical expressions, break them down into smaller parts to analyze the truth values systematically.

Explore when the compound logical statement (¬p ∧ r) ∨ (q ∧ ¬r) evaluates to false. Learn about logical operators, truth values, and conditions that affect the truth of complex logical expressions. Suitable for high school students and anyone interested in logic and boolean algebra.

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