Let’s analyze the problem and the truth table shown in the image. The task is to prove that the compound proposition $( (p \rightarrow q) \land (q \rightarrow r) ) \rightarrow (p \rightarrow r)$ is a tautology, meaning it is always true.

1. Logical Breakdown of the Proposition

• The compound proposition involves:
  1. $p \rightarrow q$ (if $p$, then $q$).
  2. $q \rightarrow r$ (if $q$, then $r$).
  3. Combine the above using $\land$: $(p \rightarrow q) \land (q \rightarrow r)$.
  4. Compare this to $p \rightarrow r$: $((p \rightarrow q) \land (q \rightarrow r)) \rightarrow (p \rightarrow r)$.

The truth table is used to verify if this statement is true for all possible truth values of $p$, $q$, and $r$.

2. Structure of the Truth Table

From the image, the truth table calculates the following columns:

• $p$, $q$, and $r$: All possible truth value combinations for these variables.
• $p \rightarrow q$: Derived using the implication rule ($p \rightarrow q$ is false only when $p = T$ and $q = F$).
• $q \rightarrow r$: Similar logic as above.
• $(p \rightarrow q) \land (q \rightarrow r)$: Logical AND of the above two columns.
• $p \rightarrow r$: Direct implication rule.
• $[(p \rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r)$: Final column checking if the compound proposition holds.

3. Correctness of the Truth Table

• Observation: The last column of the table in the image contains only $T$ (true) for all possible combinations of $p$, $q$, and $r$.
• Reasoning: This indicates that the proposition $( (p \rightarrow q) \land (q \rightarrow r) ) \rightarrow (p \rightarrow r)$ is always true, regardless of the truth values of $p$, $q$, or $r$.

4. Conclusion

The truth table is correct. The final column confirms that the given compound proposition is indeed a tautology.

Would you like further clarification on how to construct truth tables or explain any specific step in more detail?

Related Questions:

1. What is the definition of a tautology in logic?
2. How does the implication ($p \rightarrow q$) work in truth tables?
3. Can you construct a truth table for $(p \lor q) \rightarrow r$?
4. What happens if any row in the last column is false? Would it still be a tautology?
5. How does this concept apply to practical reasoning or programming logic?

Tip:

When constructing truth tables for logical propositions, make sure to calculate step-by-step and include intermediate columns to avoid errors.