The problem is asking to identify which expressions are equivalent to $p \land (q \lor r)$. Let's analyze each option:

1. $(p \lor q) \land (p \lor r)$: This is the distributive form of $p \land (q \lor r)$, because: $p \land (q \lor r) \equiv (p \lor q) \land (p \lor r)$ So, this option is equivalent.

2. $(q \lor r) \land p$: This is just a rearrangement of the original expression due to the commutative property of conjunction ($\land$), which states that $A \land B \equiv B \land A$. Therefore, this is also equivalent.

3. $\neg p \lor (\neg q \land \neg r)$: This is not equivalent. It resembles De Morgan’s Law but applies to a different structure.

4. $\neg p \lor (q \lor r)$: This is not equivalent because it changes the original structure of the expression. It resembles a negation but is not the correct transformation.

5. $(p \land q) \lor (p \land r)$: This is another valid form due to the distributive property. It’s the same as option 1, but written differently. Therefore, this is equivalent.

Final Answer:

The equivalent expressions are:

• $(p \lor q) \land (p \lor r)$
• $(q \lor r) \land p$
• $(p \land q) \lor (p \land r)$

Would you like any further details or have additional questions?

Here are 5 related questions for further exploration:

1. What is the commutative property in logic?
2. How does De Morgan's Law work for negations of conjunctions and disjunctions?
3. What is the distributive property in logical expressions?
4. Can truth tables be used to verify logical equivalence?
5. How do we simplify complex logical expressions using Boolean algebra?

Tip: When unsure if two expressions are equivalent, using a truth table can be an effective way to compare their truth values across all possible inputs.