Problem 5 - Discrete Mathematics with Applications

Question

Determine if each is a boolean expression, where each variable is boolean. $$\left(\left(x y^{\prime}\right)^{\prime}\right)^{\prime}$$

Step-by-Step Solution

Answer

The simplified expression \(x \cdot y^{\prime}\) consists of boolean variables (\(x\) and \(y^{\prime}\)) combined with boolean operators (the NOT operator on \(y\) and the AND operator between \(x\) and \(y^{\prime}\)). Therefore, it is a boolean expression.

Step 1: Understand the notation

In this expression, a prime symbol (\(\prime\)) represents the NOT operator. Given a boolean value \(A\), the NOT operator reverses the value of the variable. For example, \(A^{\prime}\) would be true (1) if \(A\) is false (0) and vice versa.

Step 2: Simplify the expression

Let's simplify the given expression step by step: Given expression: \[\left(\left(x y^{\prime}\right)^{\prime}\right)^{\prime}\] Step 2.1: Apply De Morgan's law: De Morgan's law states that the negation of a conjunction is the disjunction of the negations: \[(A \cdot B)^{\prime} = A^{\prime} + B^{\prime}\] Using De Morgan's law on the inner part of the expression \(\left(x y^{\prime}\right)^{\prime}\), we get: \[\left(x^{\prime} + \left(y^{\prime}\right)^{\prime}\right)\] Step 2.2: Simplify the double prime on \(y\): The double prime (\(\prime\prime\)) in the expression \(\left(y^{\prime}\right)^{\prime}\) is the same as applying the NOT operator twice, which results in the original value of the variable. Therefore, \(\left(y^{\prime}\right)^{\prime} = y\). The expression now becomes: \[\left(x^{\prime} + y\right)\] Step 2.3: Apply De Morgan's law on the expression: As we did in Step 2.1, now apply De Morgan's law on the outer part of the expression: \[\left(\left(x^{\prime} + y\right)^{\prime}\right) = x^{\prime\prime} \cdot y^{\prime}\] Step 2.4: Simplify the double prime on \(x\): As we did in Step 2.2, the double prime (\(\prime\prime\)) in the expression \(x^{\prime\prime}\) is same as applying the NOT operator twice, which results in the original value of the variable. Therefore, \(x^{\prime\prime} = x\), and the final simplified expression becomes: \[x \cdot y^{\prime}\]

Step 3: Determine if the simplified expression is a boolean expression

The simplified expression \(x \cdot y^{\prime}\) consists of boolean variables (\(x\) and \(y^{\prime}\)) combined with boolean operators (the NOT operator on \(y\) and the AND operator between \(x\) and \(y^{\prime}\)). Therefore, the expression is indeed a boolean expression.

Key Concepts

De Morgan's law

De Morgan's laws are essential rules in Boolean algebra that help in simplifying expressions involving logical operations. These laws are named after the British mathematician Augustus De Morgan. They describe the relationship between the `AND (⋅)` and `OR (+)` operators through negation.

There are two primary versions of De Morgan's laws:

• Negation of Conjunction: \[(A \cdot B)' = A' + B'\]
  This states that the negation of two variables being true (AND operation) is the same as either of the variables being false (OR operation).
• Negation of Disjunction: \[(A + B)' = A' \cdot B'\]
  Here, the negation of either or both variables being true (OR operation) is equivalent to each of them being false (AND operation).

Understanding and applying De Morgan's laws is fundamental in simplifying and transforming Boolean expressions to make complex logical operations more manageable.

Boolean expression

In digital logic and computer science, a Boolean expression is a combination of Boolean variables and operators that yield a Boolean result, i.e., 0 (false) or 1 (true).

Boolean expressions can include:

• Boolean Variables: Typically represented by letters like x, y, A, or B, which can either be true (1) or false (0).
• Boolean Operators: The most common operators are AND (⋅), OR (+), and NOT (').

Boolean expressions are widely used in areas such as:

• Digital Circuit Design: Describing the logic gates and their functions in an electronic device.
• Algorithm Design: Representing conditions and logic flows within programs.

The ability to simplify Boolean expressions, using principles like De Morgan's laws or other algebraic techniques, is crucial for optimizing computational logic in these applications.

NOT operator

The NOT operator is one of the fundamental operations in Boolean algebra. It is used to invert the value of a Boolean variable or expression.

Here's how it works:

• Principle: If you have a Boolean variable A, the expression A' implies NOT A. This operation changes true (1) to false (0) and vice versa.
• Double NEGATION: When NOT is applied twice, as in A'' (A prime-prime), the original value is restored: \[A'' = A\].

In practical terms, the NOT operator is extensively used in:

• Logic gates: To create inverters that change input signals in digital circuits.
• Logical conditions: In programming, where decision-making often involves checking NOT conditions.

The understanding of how NOT operates, especially when combined with other operators, helps in the manipulation and simplification of complex Boolean expressions.