Problem 42 - Discrete Mathematics with Applications

Question

Use De Morgan's laws to verify each. (Hint: \(p \rightarrow q \equiv \sim p \vee q\) ). $$\sim(\sim p \wedge \sim q) \equiv p \vee q$$

Step-by-Step Solution

Answer

By applying De Morgan's law to the given expression \(\sim(\sim p \wedge \sim q)\), we get \(\sim \sim p \vee \sim \sim q\), which simplifies to \(p \vee q\) using the double negation law. Thus, the given equivalence \(\sim(\sim p \wedge \sim q) \equiv p \vee q\) is verified using De Morgan's laws.

Step 1: Write down the given equivalence and De Morgan's laws

We are given the following equivalence to verify: \[\sim(\sim p \wedge \sim q) \equiv p \vee q\] De Morgan's laws state that: \(\sim(p \wedge q) \equiv \sim p \vee \sim q\) \(\sim(p \vee q) \equiv \sim p \wedge \sim q\)

Step 2: Apply De Morgan's law to the given expression

Let's apply De Morgan's law to the left side of the given equivalence: \[\sim(\sim p \wedge \sim q) = \sim \sim p \vee \sim \sim q\] By applying the double negation law, we get: \[p \vee q \] Now we can see that the simplified expression is equal to the right side of the equivalence: \[\sim(\sim p \wedge \sim q) \equiv p \vee q\] Therefore, we have verified the given equivalence using De Morgan's laws.

Key Concepts

Propositional Logic

Propositional logic is a fundamental branch of logic that deals with propositions and their relationships. Propositions are simple statements that can be either true or false. In propositional logic, these propositions are usually represented by variables, such as \( p \), \( q \), etc. The power of propositional logic stems from its ability to form intricate logical expressions using connectives such as:

• \( \land \) (and)
• \( \lor \) (or)
• \( \rightarrow \) (implies)
• \( \sim \) (not)

These connectives allow us to explore how different propositions relate to each other. For instance, in the given exercise, De Morgan's laws serve as crucial tools for transitioning between these logical expressions.Understanding propositional logic is essential for many fields, including computer science, mathematics, and philosophy, as it helps to develop rigorous thinking skills.

Double Negation

Double negation is a simple yet powerful concept in propositional logic. It involves the idea that applying negation twice to a proposition returns it to its original state. This rule is expressed as:i.e., \( \sim \sim p \equiv p \)This principle is widely used in logical derivations to simplify expressions. It can be especially helpful when applied within the context of other logical laws, such as De Morgan's laws.In the exercise, double negation is applied to transform \( \sim(\sim p \land \sim q) \) into \( \sim \sim p \lor \sim \sim q \), which simplifies to \( p \lor q \). This step demonstrates how double negation helps to reduce complex expressions to easier forms that are logically equivalent to the original.

Logical Equivalence

Logical equivalence is a fundamental concept in propositional logic that signifies that two expressions evaluate to the same truth value in every possible scenario. Two expressions are logically equivalent if they are true or false under the same conditions, which is denoted by the symbol \( \equiv \).Logical equivalence is useful for transforming and simplifying logical expressions while preserving their truth. In the given solution, the goal was to show that \( \sim(\sim p \wedge \sim q) \equiv p \vee q \), demonstrating their equivalence through logical laws.

How to Prove Logical Equivalence

- Use established laws such as De Morgan's laws, double negation, and distribution.- Simplify both sides of the equivalence step by step, ensuring each transformation is valid.- Confirm that the simplified expressions on both sides are identical.By proving logical equivalence, we can better understand logical structures and reason about them effectively, which is invaluable in tasks like mathematical proof writing and algorithm optimization.