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:
Where:
- is the negation of (true if is false and vice versa).
- denotes "and."
- denotes "or."
- is the negation of .
Let's evaluate the compound statement under the given conditions.
1. is true, is true, is true
- false
- false
This condition makes the statement FALSE.
2. is false, is true, is true
- true
- false
This condition makes the statement TRUE.
3. is true, is true, is false
- false
- true
This condition makes the statement TRUE.
4. is false, is false, is true
- true
- false
This condition makes the statement TRUE.
Conclusion:
The compound statement is FALSE only under the first condition:
- is true, is true, is true.
Would you like further details or have any questions?
Here are five related questions:
- How would the truth value change if both and were false?
- What is the truth value if is false, is false, and is false?
- How does the distributive property of logic apply to this expression?
- What would the result be if the expression used instead of inside the parentheses?
- How would adding another logical term, like , affect the statement?
Tip: When evaluating complex logical expressions, break them down into smaller parts to analyze the truth values systematically.
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Math Problem Analysis
Mathematical Concepts
Logic
Boolean Algebra
Formulas
-
Theorems
-
Suitable Grade Level
High School