We want to simplify the logical expression $\left( (p \lor \neg q) \land (\neg p \lor \neg q) \right) \lor q$ without using truth tables. Let's go through it step-by-step.

Step 1: Distribute and simplify the conjunction

We start by distributing $\land$ over $\lor$ in the first part of the expression.

$(p \lor \neg q) \land (\neg p \lor \neg q)$

We apply the distributive property of $\land$ over $\lor$:

$= (p \land \neg p) \lor (p \land \neg q) \lor (\neg q \land \neg p) \lor (\neg q \land \neg q)$

Simplify the terms:

• $p \land \neg p = \text{false}$ (since $p$ and $\neg p$ cannot both be true).
• $\neg q \land \neg q = \neg q$ (since $\neg q$ is true whenever $q$ is false).

Thus, we are left with:

$= (p \land \neg q) \lor (\neg p \land \neg q) \lor \neg q$

Step 2: Factor out $\neg q$

Notice that all the terms in the simplified expression involve $\neg q$, so we can factor out $\neg q$:

$= \neg q \land (p \lor \neg p \lor \neg q)$

Since $p \lor \neg p$ is a tautology (i.e., always true), the expression simplifies to:

$= \neg q \land \text{true} = \neg q$

So, the expression $(p \lor \neg q) \land (\neg p \lor \neg q)$ simplifies to $\neg q$.

Step 3: Incorporate the $\lor q$

Now, we go back to the original expression:

$\left( (p \lor \neg q) \land (\neg p \lor \neg q) \right) \lor q$

Substitute the simplified form $\neg q$ for the conjunction:

$= \neg q \lor q$

Step 4: Apply the Tautology

The expression $\neg q \lor q$ is a tautology, because it is always true, regardless of the value of $q$:

$= \text{true}$

Final Result:

Thus, the original expression simplifies to:

$\boxed{\text{true}}$

Would you like more details or further explanation?

Here are some related questions you might find interesting:

1. What are the properties of logical tautologies and contradictions?
2. How do distributive, associative, and commutative properties work in logic?
3. Can you simplify the expression $(p \lor q) \land (\neg p \lor \neg q)$?
4. How would you use De Morgan's laws to simplify logical expressions?
5. What are some common logical equivalences used in propositional logic?

Tip: In propositional logic, always check for tautologies (statements that are always true) or contradictions (statements that are always false) early in the simplification process, as they can significantly reduce the complexity of the expression.