Math Problem Statement

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

Solution

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

(¬pr)(q¬r)(\neg p \land r) \lor (q \land \neg r)

Where:

  • ¬p\neg p is the negation of pp (true if pp is false and vice versa).
  • \land denotes "and."
  • \lor denotes "or."
  • ¬r\neg r is the negation of rr.

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

1. pp is true, qq is true, rr is true

  • ¬p=\neg p = false
  • ¬r=\neg r = false

(falsetrue)(truefalse)=falsefalse=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. pp is false, qq is true, rr is true

  • ¬p=\neg p = true
  • ¬r=\neg r = false

(truetrue)(truefalse)=truefalse=true(\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. pp is true, qq is true, rr is false

  • ¬p=\neg p = false
  • ¬r=\neg r = true

(falsefalse)(truetrue)=falsetrue=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. pp is false, qq is false, rr is true

  • ¬p=\neg p = true
  • ¬r=\neg r = false

(truetrue)(falsefalse)=truefalse=true(\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:

  • pp is true, qq is true, rr 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 pp and qq were false?
  2. What is the truth value if pp is false, qq is false, and rr 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 ¬q\neg q, affect the statement?

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

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logic
Boolean Algebra

Formulas

-

Theorems

-

Suitable Grade Level

High School