Problem 45 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

Using De Morgan's laws, we can simplify the given expression as follows: 1. Applying the first De Morgan's law to \(\sim(p \wedge \sim q)\), we get \(\sim p \vee \sim (\sim q)\). 2. Using the double negation law, we simplify further to \(\sim p \vee q\). Thus, the statement \(\sim(p \wedge \sim q) = \sim p \vee q\) is verified using De Morgan's laws.

Step 1: Understand De Morgan's laws

De Morgan's laws consist of two rules that can be used to simplify logical expressions. They are: 1. \(\sim(p \wedge q) = \sim p \vee \sim q\) 2. \(\sim(p \vee q) = \sim p \wedge \sim q\)

Step 2: Apply De Morgan's laws to the left-hand side of the statement

We are given the expression \(\sim(p \wedge \sim q)\) on the left side of the statement. Using the first De Morgan's law, we can rewrite the left-hand side expression as follows: \(\sim(p \wedge \sim q) = \sim p \vee \sim (\sim q)\)

Step 3: Simplify the expression further

In the previous step, we obtained the expression \(\sim p \vee \sim (\sim q)\). Let's simplify it using the double negation law, which states that \(\sim (\sim x) = x\). Therefore, the simplified expression becomes: \(\sim p \vee \sim (\sim q) = \sim p \vee q\)

Step 4: Verify the equivalence of the expressions

Now, let's check if the simplified expression we obtained in step 3, \(\sim p \vee q\), is equal to the right-hand side of the statement. Comparing the two, we can see they are indeed equal: \(\sim(p \wedge \sim q) = \sim p \vee q\) Thus, we have verified the given statement using De Morgan's laws.

Key Concepts

Logical Expressions

Logical expressions are foundational elements in logic and mathematics. They are statements that can either be true or false. Logical expressions are often made up of variables and logical operators such as \( \wedge \) (AND), \( \vee \) (OR), and \( \sim \) (NOT). These operators help us describe the relationships between different conditions.

For instance, in the expression \( p \wedge q \), both \( p \) and \( q \) need to be true for the entire expression to be true. Conversely, in \( p \vee q \), the expression is true if at least one of \( p \) or \( q \) is true.

Understanding logical expressions is crucial because they serve as the building blocks for more complex logical statements and proofs. De Morgan's laws, as used in the original exercise, help us transform these logical expressions into equivalent forms, making them easier to analyze and solve.

Double Negation Law

The double negation law is a simple yet powerful principle in logic. It states that a double negation of a proposition is logically equivalent to the proposition itself. In mathematical terms, \( \sim (\sim p) = p \).

• This law reflects the idea that negating a truth statement twice will give you back the original statement.
• Utilizing the double negation law simplifies logical expressions and helps in achieving logical equivalence.

For example, in the step-by-step solution of the exercise, \( \sim (\sim q) = q \) was used to simplify the expression \( \sim p \vee \sim (\sim q) \) to \( \sim p \vee q \).

This simplification is crucial for verifying the equivalence of logical expressions, as it provides a way to directly compare expressions more easily.

Logical Equivalence

Logical equivalence is a fundamental concept in logic. Two expressions are said to be logically equivalent if they have the same truth values in every possible scenario. This means whenever one expression is true, the other is also true, and vice versa.

Logical equivalence is denoted by the symbol \( \equiv \). For instance, using De Morgan's laws and the double negation law, we can prove logical equivalences between different expressions.

• They provide a method to transform complex logical statements into simpler forms without changing their meaning.
• This allows easier reasoning and verification of logical conclusions.

In the original exercise, the logical equivalence \( \sim(p \wedge \sim q) \equiv \sim p \vee q \) was established using De Morgan's law.

Here, understanding logical equivalence helps ensure that transformations and simplifications of logical expressions preserve their original logical meanings.