Prove the given expression is a tautology by developing a series of logical equivalence to demonstrate that it is logically equivalent to T.

In logic, a **tautology** is a compound statement that is always true, regardless of the truth values of the individual statements involved. To prove an expression is a tautology, we must develop logical equivalences step-by-step to show that it always simplifies to **T (true)**.

However, since you didn't provide the specific expression that needs to be proved as a tautology, I will outline a general approach to proving any logical expression is a tautology using logical equivalences. Once you provide the specific expression, I can apply this method to it directly.

### Steps to Prove a Logical Expression is a Tautology

1. **Write the given logical expression**: Express the compound logical expression using logical operators like \(\land\) (and), \(\lor\) (or), \(\neg\) (not), \(\rightarrow\) (implies), \(\leftrightarrow\) (if and only if).

2. **Simplify using logical equivalences**: Apply the following commonly used logical equivalences to simplify the expression:
   - **Double Negation**: \(\neg(\neg P) \equiv P\)
   - **De Morgan's Laws**: 
     - \(\neg(P \land Q) \equiv (\neg P \lor \neg Q)\)
     - \(\neg(P \lor Q) \equiv (\neg P \land \neg Q)\)
   - **Implication Equivalence**: 
     - \(P \rightarrow Q \equiv \neg P \lor Q\)
   - **Contrapositive**: 
     - \(P \rightarrow Q \equiv \neg Q \rightarrow \neg P\)
   - **Commutative, Associative, and Distributive Laws**: These apply similar to algebra.
   - **Identity Laws**: 
     - \(P \land T \equiv P\)
     - \(P \lor F \equiv P\)
   - **Domination Laws**: 
     - \(P \lor T \equiv T\)
     - \(P \land F \equiv F\)
   - **Idempotent Laws**: 
     - \(P \land P \equiv P\)
     - \(P \lor P \equiv P\)
   - **Negation Laws**: 
     - \(P \lor \neg P \equiv T\)
     - \(P \land \neg P \equiv F\)

3. **Check for truth values**: After simplifying, if the expression reduces to \(T\) (true) for all possible truth values of the components, then the expression is a tautology.

4. **Use truth tables (optional)**: If needed, construct a truth table for the expression to confirm that the expression is true in all cases. If every row in the truth table results in true (\(T\)), the expression is a tautology.

### Example:

Consider the expression \( (P \lor \neg P) \).

1. This is a well-known tautology, but let's prove it step-by-step.
2. Apply the **Negation Law**:
   \[
   P \lor \neg P \equiv T
   \]
3. Since this simplifies directly to \(T\), the expression is a tautology.

Please provide the specific logical expression you'd like to prove as a tautology, and I will guide you through the solution!

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### 5 Follow-up Questions:
1. Would you like to provide a specific logical expression to prove as a tautology?
2. Should I explain any particular logical equivalence in more detail?
3. Would you like to see an example using truth tables?
4. Do you need clarification on any of the laws I mentioned?
5. Are there specific types of logical expressions (e.g., implications, conjunctions) you'd like to explore further?

### Tip:
When proving tautologies, try simplifying using De Morgan's laws and implication equivalences early to make the expression easier to manage.

Learn how to prove that a logical expression is a tautology by applying a series of logical equivalences. This guide covers core concepts such as double negation, De Morgan's Laws, implication equivalence, and negation laws, with examples suitable for introductory college-level logic students.

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