Problem 3 - Discrete Mathematics with Applications

Question

Simplify the boolean expression represented by each Karnaugh map. $$\begin{aligned}&\begin{array}{lllll}\qquad yz \quad y z^{\prime} \quad y^{\prime} z^{\prime}\quad y^{\prime}z \end{array} \\ &\begin{array}{lllll}wx \\\ wx^{\prime} \\ w^{\prime} x^{\prime}\\\ w^{\prime} x\end{array} \quad\begin{array}{|c|c|c|c|}\hline & & 1 & 1 \\\\\hline & & d & d \\\\\hline d & d & d & d \\\\\hline 1 & 1 & d & d \\\\\hline\end{array}\end{aligned}$$

Step-by-Step Solution

Answer

The simplified boolean expression for the given Karnaugh map is \(y' z + w' z + w' x'\).

Step 1: Identify the groups

In this Karnaugh map, we can identify two groups of 1s and one group of don't-care (d) terms in such a way that it either covers each 1 or d once or overlaps with a 1. The groups are: - Group 1: The two 1s in the top row - Group 2: The four 1s in the last column - Group 3: All the don't-care (d) terms

Step 2: Write the Sum of Products expression for each group

Now, we will find the Sum of Products (SOP) expression for each group: - Group 1: \(y' z\) - Group 2: \(w' z\) - Group 3: \(w' x'\)

Step 3: Combine the expressions using OR operation

To simplify the boolean expression, we will combine the obtained expressions from Step 2 using the OR operation. The simplified boolean expression is given by: Simplified Boolean Expression = \(y' z + w' z + w' x'\)

Key Concepts

Boolean Expression

A Boolean expression is a mathematical way of representing logic in terms of symbolic operations. These operations include AND, OR, and NOT, which are fundamental in digital electronics. Boolean expressions are used to design and simplify digital circuits.
For example, in this exercise, we're looking at a Karnaugh map that has been converted into a Boolean expression. The symbols such as 1s and 0s indicate the presence or absence of a condition. The Karnaugh map is a tool that visually represents these conditions to help simplify Boolean expressions.

• 1 represents a "true" condition.
• 0 represents a "false" condition.
• "d" or "don't-care" conditions are placeholders that can be either 1 or 0 to aid in simplification.

Understanding the initial Boolean expression is the first step towards simplifying it and finding the most efficient way to represent a digital circuit.

Sum of Products

The Sum of Products (SOP) is a form of expression used in the simplification of Boolean logic. It consists of the logical OR operation applied to one or more logical AND operations.
In the context of this exercise, the SOP expression is derived from the groups found in the Karnaugh map.

• Each "product" is a term in the expression where variables are ANDed together.
• These products are then ORed to form a complete expression that describes the logic of the Karnaugh map.

To write the SOP expression for each group, consider:- Group 1 formed the SOP term: \(y' z\)- Group 2 formed the SOP term: \(w' z\)- Group 3 formed the SOP term: \(w' x'\)These are then combined using the OR operation to develop the final SOP expression \(y' z + w' z + w' x'\). This gives a more specific logical description of the electronic circuit.

Simplification

Simplification in Boolean algebra refers to the process of reducing a complex expression to its simplest form. The goal is to minimize the number of terms and operations, which directly correlates to fewer logic gates in a digital circuit.
With this exercise's Karnaugh map, the simplification involves recognizing groups of adjacent 1s and using them to form simpler expressions. This includes incorporating don't-care terms strategically to make the expression as minimal as possible.

• Groups are created within the Karnaugh map to help identify patterns.
• The objective is to cover all the "true" conditions (1s) using the fewest terms.
• This reduces the complexity and cost of the corresponding digital circuit.

The simplified expression \(y' z + w' z + w' x'\) reflects a more efficient logic design, as redundant terms are eliminated.

Logic Gates

Logic gates are the building blocks of digital circuits. Each gate represents a basic operation of Boolean algebra. In digital design, logic gates perform the computations defined by a Boolean expression.
In the simplified Boolean expression \(y' z + w' z + w' x'\), different types of gates are used:

• AND gates implement the "product" operations like \(y' z\) or \(w' z\).
• OR gates are used to combine these products into the overall "sum," as represented by the + sign.
• NOT gates (also called inverters) are used to express the negation seen in \(y'\) or \(w'\).

Constructing logic circuits using gates directly from the simplified expression ensures that the resulting electronic design is efficient and cost-effective. This exercise showcases how a Karnaugh map can provide insights into the optimal arrangement of logic gates.