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sederhanakan dengan menggunakan hukum logika

Let's work through simplifying these logical expressions using the rules of logic step by step.

### (a) \( p \lor (p \land (p \lor q)) \equiv p \)

1. Apply **Absorption Law**: 
   \[
   p \lor (p \land q) \equiv p
   \]
   This simplifies the inner expression:
   \[
   p \lor (p \land (p \lor q)) \equiv p \lor p \equiv p
   \]
   Therefore, the simplified form is:
   \[
   p \equiv p
   \]

### (b) \(\neg((p \lor q) \land r) \lor \neg q \equiv q \land r\)

1. Apply **De Morgan's Law** on the negation:
   \[
   \neg((p \lor q) \land r) \equiv \neg(p \lor q) \lor \neg r \equiv (\neg p \land \neg q) \lor \neg r
   \]
   The full expression becomes:
   \[
   (\neg p \land \neg q) \lor \neg r \lor \neg q \equiv \neg p \lor \neg r \lor \neg q
   \]

2. Now compare both sides, but we can’t simplify it directly to \(q \land r\) by basic logic laws alone. We’ll need additional steps (such as truth tables) to verify equivalence for this expression.

### (c) \( p \lor q \lor (\neg p \land \neg q \land r) \equiv p \lor q \lor r \)

1. Distribute the first part using **Distributive Law**:
   \[
   (p \lor q) \lor (\neg p \land \neg q \land r)
   \]
2. Apply **Absorption Law** to simplify:
   \[
   p \lor q \lor r
   \]
   Therefore, the simplified form is:
   \[
   p \lor q \lor r
   \]

### (d) \( (\neg(p \lor q) \Rightarrow (p \land q \land r)) \equiv p \land q \)

1. Express the implication using the **Implication Law**:
   \[
   \neg(p \lor q) \Rightarrow (p \land q \land r) \equiv (\neg \neg(p \lor q)) \lor (p \land q \land r)
   \equiv (p \lor q) \lor (p \land q \land r)
   \]
2. Apply **Absorption Law**:
   \[
   (p \lor q) \equiv p \land q
   \]
   Therefore, the simplified form is:
   \[
   p \land q
   \]

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These are the simplified results using logical laws. Would you like further clarifications or details on any step?

Here are 5 related questions:
1. Can you list the basic logical equivalence laws?
2. How can De Morgan's laws simplify complex expressions?
3. Why does absorption simplify \( p \lor (p \land q) \) to \( p \)?
4. What is the difference between logical implication and biconditional?
5. How can truth tables verify logical equivalences?

**Tip:** When simplifying logical expressions, always start by identifying which laws (like De Morgan's or distribution) directly apply to inner expressions.

Sederhanakan dengan menggunakan hukum logika.

Simplify logical expressions step by step using Boolean laws like De Morgan's Theorem, Absorption Law, and Distributive Law. This guide is suitable for advanced high school or college-level students looking to understand propositional logic.

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