To simplify the Boolean expression:

$¬A ∨ B ∨ C ∨ D ∧ ¬B ∨ ¬C ∧ ¬D ∨ A ∧ ¬A ∨ C$

Let's break it down step by step, applying logical operations (like De Morgan’s Law, distributive property, etc.) where appropriate:

Step 1: Group terms based on precedence

Boolean logic follows the order of operations where:

• $\neg$ (NOT) has the highest precedence,
• $\land$ (AND) comes next,
• $\lor$ (OR) has the lowest precedence.

So we first need to group the expression properly.

$(¬A ∨ B ∨ C ∨ (D ∧ ¬B) ∨ (¬C ∧ ¬D) ∨ (A ∧ ¬A) ∨ C)$

Step 2: Simplify internal terms

• $A \land ¬A$ is always false (contradiction): $A \land ¬A = 0$

So the expression becomes: $(¬A ∨ B ∨ C ∨ (D ∧ ¬B) ∨ (¬C ∧ ¬D) ∨ C)$

Step 3: Remove redundant terms

• $C \lor C = C$ (Idempotent Law): $¬A ∨ B ∨ C ∨ (D ∧ ¬B) ∨ (¬C ∧ ¬D)$

Step 4: Simplify further

• $¬A \lor A = 1$ (Tautology)
  • Since we don't have this directly anymore, proceed with combining terms.

Now, the expression $(¬A ∨ B ∨ C ∨ (D ∧ ¬B) ∨ (¬C ∧ ¬D))$ cannot be simplified further without additional context or assumption of specific truth values.

Final Simplified Expression:

$¬A ∨ B ∨ C ∨ (D ∧ ¬B) ∨ (¬C ∧ ¬D)$

This is the simplified form of the Boolean expression.

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Would you like further details or explanations on any specific step?

Relative questions:

1. How do Boolean operations follow precedence rules?
2. What is De Morgan's Law in Boolean algebra?
3. Can you simplify more complex Boolean expressions with multiple negations?
4. How does idempotent law apply in Boolean logic?
5. What other laws in Boolean algebra are commonly used in simplification?

Tip:

When simplifying Boolean expressions, always apply the precedence rules first (negation, conjunction, then disjunction), and look for patterns like contradictions and tautologies to simplify terms faster.