Consider the propositional formula  ( NOT ( P )  IFF  Q )  IMPLIES  R  and the formula  ( P  AND  Q )  OR  ( NOT ( P )  AND  NOT ( Q ) )  OR  R .Prove that the two formulas are equivalent, using logical rules.

We are tasked with proving that the two formulas are logically equivalent. The two formulas are:

1. \[ ( \neg P \iff Q ) \implies R \]
2. \[ ( P \land Q ) \lor ( \neg P \land \neg Q ) \lor R \]

To show that these two are equivalent, we will simplify both formulas step by step using logical rules.

### Formula 1: \(( \neg P \iff Q ) \implies R\)

#### Step 1: Expand the biconditional (IFF)

The biconditional \(\neg P \iff Q\) means \((\neg P \implies Q) \land (Q \implies \neg P)\). So, the first formula becomes:

\[
( (\neg P \implies Q) \land (Q \implies \neg P) ) \implies R
\]

#### Step 2: Express implications in terms of disjunction

Using the implication rule \((A \implies B) \equiv (\neg A \lor B)\), we rewrite both implications:

- \(\neg P \implies Q \equiv P \lor Q\)
- \(Q \implies \neg P \equiv \neg Q \lor \neg P\)

Thus, the formula becomes:

\[
((P \lor Q) \land (\neg Q \lor \neg P)) \implies R
\]

#### Step 3: Simplify the conjunction

Let's analyze the conjunction \((P \lor Q) \land (\neg Q \lor \neg P)\). This is a form of the **exclusive OR** (\(P \oplus Q\)), because it says that either \(P\) is true and \(Q\) is false,

Consider the propositional formula (NOT (P) IFF Q) IMPLIES R and the formula (P AND Q) OR (NOT (P) AND NOT (Q)) OR R. Prove that the two formulas are equivalent, using logical rules.

This problem demonstrates how to prove the logical equivalence between two propositional formulas using logical rules. Concepts such as biconditional (IFF), implication (IMPLIES), and De Morgan's laws are applied step-by-step to simplify and compare the expressions. This exercise is ideal for students studying discrete mathematics or logic at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation