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The proposition provided is a complex logical formula, and the task is to determine how many false values appear in the main matrix (truth table) for the expression.

### Expression:
\[
[\neg p \lor (r \rightarrow \neg q)] \leftrightarrow [(\neg q \land p) \Delta \neg r]
\]

### Steps to solve:
1. **Variables**: We have three variables, \(p\), \(q\), and \(r\). Since these are binary logical variables, there will be \(2^3 = 8\) combinations of truth values.

2. **Construct the Truth Table**: We need to evaluate each component of the expression for every possible combination of \(p\), \(q\), and \(r\). Key logical operations include:
   - Negation (\(\neg\))
   - Logical OR (\(\lor\))
   - Logical AND (\(\land\))
   - Implication (\(\rightarrow\))
   - Exclusive OR (\(\Delta\))
   - Biconditional (\(\leftrightarrow\))

3. **Analyze the main expression** to create the truth table and count the false values.

Let me create the truth table to evaluate this expression and calculate the number of false values.



Luego de desarrollar la siguiente proposición compuesta, mediante una tabla de verdad; indica la cantidad de valores falsos que resulta en su matriz principal: [¬p ∨ (r → ¬q)] ↔ [(¬q ∧ p) Δ ¬r]

Learn how to solve a complex propositional logic statement using a truth table. This problem involves evaluating logical operations such as negation, conjunction, disjunction, implication, exclusive or, and biconditional equivalence. Suitable for undergraduate-level logic courses.

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