To simplify the logical expression $\neg(p \leftrightarrow q) \leftrightarrow (p \leftrightarrow \neg q)$, let's break it down step by step:

Step 1: Understand the biconditional (↔) operator

The biconditional $p \leftrightarrow q$ is true when both $p$ and $q$ have the same truth value, i.e., both are true or both are false.

It can be rewritten as: $p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q)$ Likewise, $p \leftrightarrow \neg q \equiv (p \land \neg q) \lor (\neg p \land q)$

Step 2: Analyze the negation of the first biconditional

The expression $\neg(p \leftrightarrow q)$ can be simplified. Since $p \leftrightarrow q$ means they have the same truth value, $\neg(p \leftrightarrow q)$ means they have different truth values.

So, $\neg(p \leftrightarrow q) \equiv (p \land \neg q) \lor (\neg p \land q)$ This is essentially the same as the XOR (exclusive or) operation: $p \oplus q$.

Step 3: Work on the whole expression

We now have: $(p \oplus q) \leftrightarrow (p \leftrightarrow \neg q)$ We need to check when the two sides of the biconditional are equivalent.

Step 4: Truth table approach

To see when the biconditional is true, we can use a truth table to compare $p \oplus q$ and $p \leftrightarrow \neg q$.

$p$	$q$	$p \leftrightarrow q$	$\neg(p \leftrightarrow q)$	$p \leftrightarrow \neg q$	Final expression $\neg(p \leftrightarrow q) \leftrightarrow (p \leftrightarrow \neg q)$
T	T	T	F	F	T
T	F	F	T	T	T
F	T	F	T	T	T
F	F	T	F	F	T

Step 5: Conclusion

The truth table shows that $\neg(p \leftrightarrow q) \leftrightarrow (p \leftrightarrow \neg q)$ is always true, making this a tautology.

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Would you like further details or have any questions?

5 Related Questions:

1. What is the difference between biconditional (↔) and exclusive or (XOR)?
2. How does a truth table help in proving logical equivalence?
3. Can you explain more about tautologies in logic?
4. How do we simplify complex logical expressions?
5. What are some other examples of tautologies in logic?

Tip:

When simplifying logical expressions, breaking them down into smaller parts and using truth tables can help ensure accuracy in the process!