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.