We want to prove that the two logical statements:

1. $(p \to q) \lor (q \to r)$
2. $p \to (q \lor r)$

are logically equivalent. To do this, we'll analyze both expressions using logical identities and transformations.

Step 1: Rewrite the implications

Recall that an implication $p \to q$ is logically equivalent to $\neg p \lor q$. We can apply this identity to both expressions.

Expression 1: $(p \to q) \lor (q \to r)$

Using the identity $p \to q \equiv \neg p \lor q$ on both implications:

$(p \to q) \equiv \neg p \lor q$ $(q \to r) \equiv \neg q \lor r$

Substituting into the first expression:

$(\neg p \lor q) \lor (\neg q \lor r)$

This is a disjunction of two terms: $(\neg p \lor q)$ and $(\neg q \lor r)$.

Expression 2: $p \to (q \lor r)$

Similarly, we apply the implication identity to the second expression:

$p \to (q \lor r) \equiv \neg p \lor (q \lor r)$

Step 2: Simplify both expressions

Now, we simplify both expressions to see if they are logically equivalent.

Simplifying Expression 1: $(\neg p \lor q) \lor (\neg q \lor r)$

We can use the associative and commutative properties of logical disjunction ($\lor$) to group the terms:

$(\neg p \lor q) \lor (\neg q \lor r) = \neg p \lor (q \lor (\neg q \lor r))$

We simplify the inner part. Notice that $q \lor \neg q$ is a tautology (i.e., always true):

$q \lor \neg q = \text{True}$

Therefore, the expression becomes:

$\neg p \lor \text{True} \lor r = \text{True}$

Since a disjunction with "True" is always true, we get:

$\neg p \lor (q \lor r)$

Simplifying Expression 2: $\neg p \lor (q \lor r)$

This is already in its simplest form.

Step 3: Compare the simplified forms

• Expression 1 simplifies to: $\neg p \lor (q \lor r)$
• Expression 2 is: $\neg p \lor (q \lor r)$

Since both expressions are identical, we conclude that:

$(p \to q) \lor (q \to r) \equiv p \to (q \lor r)$

Conclusion:

The two expressions $(p \to q) \lor (q \to r)$ and $p \to (q \lor r)$ are logically equivalent.

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Would you like a deeper dive into any of the steps or further explanation?

Here are 5 related questions:

1. How do truth tables prove logical equivalence?
2. What are tautologies and contradictions in logic?
3. How can we use De Morgan’s laws in proving equivalences?
4. What role do associative and commutative properties play in logic?
5. How does simplifying compound logical expressions work?

Tip: When proving logical equivalence, breaking down implications using $\neg p \lor q$ is a very useful strategy!