To prove the logical equivalence $(p \rightarrow q) \land (q \rightarrow r) = p \rightarrow r$ using a truth table, we can follow these steps:

Step 1: List All Possible Truth Values for $p$, $q$, and $r$

Since $p$, $q$, and $r$ are all propositions, each can be either true (T) or false (F). The possible combinations are:

\hline p & q & r \\ \hline T & T & T \\ T & T & F \\ T & F & T \\ T & F & F \\ F & T & T \\ F & T & F \\ F & F & T \\ F & F & F \\ \hline \end{array}$$ ### Step 2: Compute $$p \rightarrow q$$, $$q \rightarrow r$$, and $$p \rightarrow r$$ The implication $$p \rightarrow q$$ is false only when $$p$$ is true and $$q$$ is false. Similarly, for $$q \rightarrow r$$ and $$p \rightarrow r$$. $$\begin{array}{|c|c|c|c|c|c|} \hline p & q & r & p \rightarrow q & q \rightarrow r & p \rightarrow r \\ \hline T & T & T & T & T & T \\ T & T & F & T & F & F \\ T & F & T & F & T & T \\ T & F & F & F & T & F \\ F & T & T & T & T & T \\ F & T & F & T & F & T \\ F & F & T & T & T & T \\ F & F & F & T & T & T \\ \hline \end{array}$$ ### Step 3: Compute $$(p \rightarrow q) \land (q \rightarrow r)$$ The conjunction $$(p \rightarrow q) \land (q \rightarrow r)$$ is true only when both $$p \rightarrow q$$ and $$q \rightarrow r$$ are true. $$\begin{array}{|c|c|c|c|c|c|c|} \hline p & q & r & p \rightarrow q & q \rightarrow r & (p \rightarrow q) \land (q \rightarrow r) & p \rightarrow r \\ \hline T & T & T & T & T & T & T \\ T & T & F & T & F & F & F \\ T & F & T & F & T & F & T \\ T & F & F & F & T & F & F \\ F & T & T & T & T & T & T \\ F & T & F & T & F & F & T \\ F & F & T & T & T & T & T \\ F & F & F & T & T & T & T \\ \hline \end{array}$$ ### Step 4: Compare $$(p \rightarrow q) \land (q \rightarrow r)$$ and $$p \rightarrow r$$ Notice that the columns for $$(p \rightarrow q) \land (q \rightarrow r)$$ and $$p \rightarrow r$$ are identical for all possible truth values of $$p$$, $$q$$, and $$r$$. ### Conclusion: Since the truth tables for $$(p \rightarrow q) \land (q \rightarrow r)$$ and $$p \rightarrow r$$ are identical, we have proven that: $$(p \rightarrow q) \land (q \rightarrow r) \equiv p \rightarrow r$$ This shows that the expressions are logically equivalent. Would you like any further details or explanations? Here are some related questions you might find interesting: 1. What is the definition of logical equivalence? 2. How does a truth table help in proving logical equivalence? 3. Can you explain the implication $$p \rightarrow q$$ in simple terms? 4. How do conjunctions and disjunctions work in logic? 5. What are the common logical connectives used in truth tables? 6. Can you demonstrate another example of proving logical equivalence using a truth table? 7. What are some practical applications of logical equivalence? 8. How can you simplify logical expressions using Boolean algebra? **Tip:** When working with truth tables, it’s important to clearly outline all possible truth values for the variables involved to ensure accuracy.