Math Problem Statement
Make a truth table
Make a truth table
Solution
The image presents a logical expression and a partially constructed truth table. Let's first break down the columns and symbols:
- , , and are the three propositions (boolean variables).
- is a conditional statement ("if then ").
- represents the logical conjunction (" AND ").
- combines the conditional with using AND.
- is the final column, representing the implication of the full expression.
Now, I'll create a complete truth table for this setup, where , , and take all possible truth values (True = T, False = F).
I'll display the full table below with all logical operations.Here is the complete truth table for the logical expression :
\hline p & q & r & p \to q & r \land q & (p \to r) \land (r \land q) & [(p \to r) \land (r \land q)] \to p \\ \hline \text{True} & \text{True} & \text{True} & \text{True} & \text{True} & \text{True} & \text{True} \\ \text{True} & \text{True} & \text{False} & \text{True} & \text{False} & \text{False} & \text{True} \\ \text{True} & \text{False} & \text{True} & \text{False} & \text{False} & \text{False} & \text{True} \\ \text{True} & \text{False} & \text{False} & \text{False} & \text{False} & \text{False} & \text{True} \\ \text{False} & \text{True} & \text{True} & \text{True} & \text{True} & \text{True} & \text{False} \\ \text{False} & \text{True} & \text{False} & \text{True} & \text{False} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{True} & \text{True} & \text{False} & \text{False} & \text{True} \\ \text{False} & \text{False} & \text{False} & \text{True} & \text{False} & \text{False} & \text{True} \\ \hline \end{array}$$ Would you like more details on any specific step of the truth table construction? Here are some related questions: 1. What is the difference between conjunction ($$\land$$) and implication ($$\to$$)? 2. How is the truth value of $$p \to q$$ determined for different inputs? 3. Why does the final expression only evaluate to False in one row? 4. How do De Morgan's laws apply to logical expressions like this? 5. What would change if we added a negation to the entire formula? **Tip**: In a truth table, $$p \to q$$ is only false when $$p$$ is true and $$q$$ is false.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Logic
Truth Tables
Propositional Logic
Formulas
p → q (If p then q)
r ∧ q (r AND q)
(p → r) ∧ (r ∧ q) (Logical conjunction)
[(p → r) ∧ (r ∧ q)] → p (Implication)
Theorems
Truth Table Construction
Implication and Conjunction in Propositional Logic
Suitable Grade Level
Grades 10-12 or Introductory College Logic