To construct a truth table for $(r \vee q) \land (r \land p)$, we need to examine all possible truth values for the propositions $p$, $q$, and $r$. Here’s a step-by-step solution for creating the truth table.

Step-by-Step Truth Table Construction

1. Identify Variables: We have three variables $p$, $q$, and $r$, which results in $2^3 = 8$ possible combinations of truth values.
2. Column Breakdown:
   • Columns for each variable: $p$, $q$, $r$
   • Intermediate expressions: $r \vee q$, $r \land p$
   • Final expression: $(r \vee q) \land (r \land p)$

Truth Table

Explanation of Columns

• $r \vee q$: This is true if either $r$ or $q$ is true.
• $r \land p$: This is true only if both $r$ and $p$ are true.
• $(r \vee q) \land (r \land p)$: This final expression is true if both $r \vee q$ and $r \land p$ are true.

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Related Questions

1. How do we create a truth table for a larger number of variables?
2. What is the difference between conjunction ($\land$) and disjunction ($\vee$) in logic?
3. Can a truth table be simplified using logical laws?
4. How do De Morgan’s laws apply to simplifying logical expressions?
5. What are the basic steps to converting a logical expression into a truth table?

Tip

Remember that each row in a truth table represents a unique combination of truth values for each variable, covering all possibilities.