We are tasked with simplifying the Boolean expression:

$S = (A \lor \neg B) \land (B \lor \neg C) \land (\neg A \lor C) \land (A \lor B \lor C) \land (\neg A \lor \neg B \lor \neg C)$

Step-by-step simplification:

1. Examine for contradictions or tautologies:

   • First, we check if there are any contradictions or tautologies within the expression. A contradiction in a Boolean expression means that certain terms cannot be true simultaneously, whereas a tautology means a term is always true.

2. Focus on key expressions: Let's start by analyzing each pair of expressions.

   • Consider $(A \lor \neg B) \land (\neg A \lor B \lor \neg C)$.
   • $A \lor \neg B$ indicates that if $A$ is true, $B$ must be false. At the same time, the term $\neg A \lor B \lor \neg C$ suggests that if $A$ is false, either $B$ is true or $C$ is false.

   Simplifying along these lines will gradually reduce the complexity. Let me break down a few of these connections further.

The simplified Boolean expression is:

$S = \neg B \land \neg C$

This means that for the expression to be true, both $B$ and $C$ must be false.

---

Do you have any questions or would you like more details?

Here are 5 related questions to expand this:

1. How can I further verify Boolean simplifications using truth tables?
2. What is the role of De Morgan's law in Boolean simplifications?
3. Can Boolean expressions represent real-world logic circuits?
4. How do Karnaugh maps help in simplifying Boolean expressions?
5. What are some common mistakes to avoid in Boolean algebra?

Tip: When simplifying Boolean expressions, always check for contradictions (false statements) and tautologies (true statements) early on, as they can greatly simplify the process.