(PVQ)->(P<->~Q) using both truth table and laws to find pcnf and pdnf

To solve the expression \((PVQ) \to (P \leftrightarrow \neg Q)\) and find the **Principal Conjunctive Normal Form (PCNF)** and **Principal Disjunctive Normal Form (PDNF)**, we'll proceed in two steps: 
1. Using a truth table.
2. Deriving PCNF and PDNF from the truth table.

### Step 1: Truth Table for \((PVQ) \to (P \leftrightarrow \neg Q)\)

Let's first break down the expression into its components:
1. \(P\)
2. \(Q\)
3. \(P \vee Q\) (the OR operation)
4. \(\neg Q\) (negation of Q)
5. \(P \leftrightarrow \neg Q\) (the biconditional, true when both have the same truth value)
6. \((P \vee Q) \to (P \leftrightarrow \neg Q)\) (implication)

We'll construct the truth table step by step:

| \(P\) | \(Q\) | \(P \vee Q\) | \(\neg Q\) | \(P \leftrightarrow \neg Q\) | \((P \vee Q) \to (P \leftrightarrow \neg Q)\) |
|:----:|:----:|:----------:|:---------:|:---------------------:|:--------------------------------------------:|
|  T   |  T   |     T      |     F     |           F           |                      F                     |
|  T   |  F   |     T      |     T     |           T           |                      T                     |
|  F   |  T   |     T      |     F     |           T           |                      T                     |
|  F   |  F   |     F      |     T     |           F           |                      T                     |

### Step 2: Laws and Simplification
Let’s break down the logical expression using laws:
1. \(P \vee Q\) (Disjunction of \(P\) and \(Q\))
2. \(\neg Q\) (Negation of \(Q\))
3. \(P \leftrightarrow \neg Q\) is true when \(P\) and \(\neg Q\) have the same truth value, meaning it behaves like \(P \leftrightarrow \neg Q \equiv (P \wedge \neg Q) \vee (\neg P \wedge Q)\).
4. The implication \((P \vee Q) \to (P \leftrightarrow \neg Q)\) can be rewritten using the implication law as \(\neg (P \vee Q) \vee (P \leftrightarrow \neg Q)\).

This helps to understand the truth table construction.

### Step 3: PCNF and PDNF

**PCNF (Principal Conjunctive Normal Form):**

From the truth table, the PCNF is formed by looking at the rows where the result is false (only row 1):
- Row 1: \(P = T\), \(Q = T\) (negate these values to form the clause)

So the clause for row 1 is: \((\neg P \vee \neg Q)\).

Hence, the **PCNF** is: 
\[
(\neg P \vee \neg Q)
\]

**PDNF (Principal Disjunctive Normal Form):**

From the truth table, the PDNF is formed by looking at the rows where the result is true (rows 2, 3, and 4):
- Row 2: \(P = T\), \(Q = F\) (write these values directly in a conjunction)
- Row 3: \(P = F\), \(Q = T\)
- Row 4: \(P = F\), \(Q = F\)

Thus, the clauses for the true rows are:
- Row 2: \(P \wedge \neg Q\)
- Row 3: \(\neg P \wedge Q\)
- Row 4: \(\neg P \wedge \neg Q\)

Hence, the **PDNF** is:
\[
(P \wedge \neg Q) \vee (\neg P \wedge Q) \vee (\neg P \wedge \neg Q)
\]

---

Would you like more details on any part? Here are some additional questions to extend the problem:

1. How is the truth table used to identify the canonical forms?
2. What is the difference between the PDNF and PCNF?
3. How does the biconditional \((\leftrightarrow)\) operator affect the truth table?
4. Can you simplify the expression \((P \vee Q) \to (P \leftrightarrow \neg Q)\) using logical laws only?
5. What is the general process of converting an expression to CNF or DNF?

**Tip**: Remember, PCNF involves using rows where the expression is false, while PDNF uses rows where the expression is true.

(PVQ) -> (P <-> ~Q) using both truth table and laws to find pcnf and pdnf

Discover how to find the Principal Conjunctive Normal Form (PCNF) and Principal Disjunctive Normal Form (PDNF) of the logical expression (PVQ) -> (P <-> ~Q). This solution uses truth tables and logical laws such as the implication and biconditional laws. Suitable for undergraduate students studying propositional logic.

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