Let's walk through this proof step-by-step and fill in the justifications using the equivalence rules from the key provided.

The given expression to prove is:

$p \rightarrow (p \land q) \equiv ?$

Step 1: Rewrite the implication using the Definition of Implication (rule $h$)

$p \rightarrow (p \land q) \equiv \neg p \lor (p \land q)$

Justification: Definition of Implication (h).

Step 2: Apply De Morgan's Law (rule $c$) to distribute $\neg p$ over the conjunction.

$\neg p \lor (p \land q) \equiv (\neg p \lor p) \land (\neg p \lor q)$

Justification: De Morgan’s Law (c).

Step 3: Use the Tautology (rule $k$) on $\neg p \lor p$.

$(\neg p \lor p) \land (\neg p \lor q) \equiv T \land (\neg p \lor q)$

Justification: Tautology (k).

Step 4: Apply the Identity Law (rule $j$) since $T \land (\neg p \lor q) \equiv \neg p \lor q$.

$T \land (\neg p \lor q) \equiv \neg p \lor q$

Justification: Identity Law (j).

Final Step: Recognize the result matches the original expression in terms of equivalence.

Thus, we’ve proved that:

$p \rightarrow (p \land q) \equiv \neg p \lor q$

This completes the proof.

Would you like further clarifications, or do you have additional questions? Here are five related questions for deeper understanding:

1. What is the general form of De Morgan’s Law, and how can it simplify logic expressions?
2. Why does the Tautology $\neg p \lor p \equiv T$ play a crucial role in logical proofs?
3. How do Identity Laws like $T \land X \equiv X$ help reduce expressions?
4. Can the Associative Property (e) be used to rearrange terms in logical statements, and if so, where?
5. What’s the difference between an equivalence and an implication in propositional logic?

Tip: When working with logical expressions, always start by breaking down implications and applying laws like De Morgan’s or Tautology early in the process.