Problem 50 - Discrete Mathematics with Applications

Question

Using a Karnaugh map, simplify each boolean expression. $$ w x^{\prime} y z^{\prime}+w x^{\prime} y^{\prime} z^{\prime}+w^{\prime} x^{\prime} y z^{\prime}+w^{\prime} x^{\prime} y^{\prime} z^{\prime} $$

Step-by-Step Solution

Answer

The question for the short answer could be: Q: Simplify the following boolean expression using Karnaugh map: $$ w x^{\prime} y z^{\prime}+w x^{\prime} y^{\prime} z^{\prime}+w^{\prime} x^{\prime} y z^{\prime}+w^{\prime} x^{\prime} y^{\prime} z^{\prime} $$ A: The simplified boolean expression is \(w x^{\prime} + x^{\prime} z^{\prime}\).

Step 1: Label the Karnaugh map for 4-variables

Firstly, create a Karnaugh map with 4-variables (w, x, y, and z). The K-map should have 16 cells arranged in a 4x4 matrix. Label the rows and columns based on the Gray code sequence (00, 01, 11, and 10) and their corresponding variable combinations. w, x will label the rows. y, z will label the columns. The matrix would look like: ``` y'z' y'z yz yz' w'x' 1 2 3 4 w'x 5 6 7 8 wx 9 10 11 12 wx' 13 14 15 16 ```

Step 2: Plot the min terms on the Karnaugh map

Next, find the min terms represented by the given boolean expression: $$ w x^{\prime} y z^{\prime}+w x^{\prime} y^{\prime} z^{\prime}+w^{\prime} x^{\prime} y z^{\prime}+w^{\prime} x^{\prime} y^{\prime} z^{\prime} $$ The min terms are: 1) \(w x^{\prime} y z^{\prime}\) -> (13) Cell 1 (cell numbers as defined in step 1) 2) \(w x^{\prime} y^{\prime} z^{\prime}\) -> (14) Cell 2 3) \(w^{\prime} x^{\prime} y z^{\prime}\) -> (05) Cell 3 4) \(w^{\prime} x^{\prime} y^{\prime} z^{\prime}\) -> (06) Cell 4 Now, place "1"s in the respective cells on the Karnaugh map: ``` y'z' y'z yz yz' w'x' 1 0 0 1 w'x 1 0 0 0 wx 0 0 0 0 wx' 0 1 0 1 ```

Step 3: Group the 1s in the Karnaugh map

Group the 1s present in the Karnaugh map. Remember to follow the rules of grouping: groups should be rectangular-shaped, and the size of groups should be a power of 2 (i.e., 1, 2, 4, 8, ...). One group can include 1's from the top row and bottom row if they are placed at the same place vertically in the row as this means the w term can be omitted. In this K-map, we have two groups: 1) Group the 1s from cells 1, 2, 14, and 15 vertically in form of a 2x2 square: \(w x^{\prime}\) 2) Group the 1s from cells 3 and 4 horizontally: \(x^{\prime} z^{\prime}\)

Step 4: Simplify the boolean expression

Based on the groups formed in the K-map, the simplified boolean expression can be written as: $$ w x^{\prime} + x^{\prime} z^{\prime} $$ The simplified boolean expression is \(w x^{\prime} + x^{\prime} z^{\prime}\).

Key Concepts

Boolean Expression Simplification

Simplifying Boolean expressions is a fundamental skill in digital logic design, involving the reduction of complex Boolean expressions to simpler forms. This process helps in creating more efficient logical circuits with fewer gates, thus reducing power consumption and cost.

The process often involves techniques such as combining like terms, applying Boolean algebra theorems, and using tools like the Karnaugh map (K-map). Karnaugh map simplification is particularly useful for visualizing relationships between different terms in an expression. By plotting min terms in a K-map, one can easily see which can be grouped together to simplify the expression. For instance, in the given exercise, the expression \(w x' y z' + w x' y' z' + w' x' y z' + w' x' y' z'\) was reduced to \(w x' + x' z'\) by forming groups within the K-map that represent common factors. This technique is more intuitive and less error-prone compared to algebraic manipulation, especially for expressions involving three or more variables.

Min Terms in Boolean Algebra

In Boolean algebra, a min term is a product (AND) of all the variables in the function, where each variable appears once in either its true form or complemented form. Min terms are crucial because they uniquely represent each row in a truth table for a Boolean function.

When creating Karnaugh maps, each min term corresponds to a particular cell within the map. The exercise provided involves min terms such as \(w x' y z'\) and \(w' x' y' z'\), which directly translate into the presence (or absence) of a '1' in specific cells of the K-map. These min terms from the original Boolean expression were plotted in the K-map to facilitate the identification of possible simplifications. Plotting correctly according to the corresponding cells is essential—Cell 1 represents \(w' x' y' z'\), Cell 13 represents \(w x' y z'\), and so forth, based on the labeling defined through the Gray code sequence.

K-map Grouping Rules

Grouping in Karnaugh maps is guided by specific rules designed to maximize simplification of the Boolean expression. Here are the key ones:

• Groups must be rectangular and contain only 1s.
• The number of 1s in a group must be a power of 2 (e.g., 1, 2, 4, 8).
• Groups should be made as large as possible to encompass the most 1s and thus simplify the expression further.
• Each group's size and shape must align with the K-map's grid structure, usually forming shapes like single cells, rows, columns, or rectangles.
• Groups can overlap if it allows for larger groupings.
• Groups can also 'wrap around' the K-map, meaning a group can connect the top and bottom rows or the leftmost and rightmost columns, to account for 'don't care' conditions.

The goal of these grouping rules is to eliminate variables. When you can group four 1s in a 2x2 square, for example, you eliminate two variables. In the exercise, two groups were made: the group of \(w x'\) from cells 1, 2, 14, 15, and the group of \(x' z'\) from cells 3 and 4. This resulted in a simplified expression of \(w x' + x' z'\), which is notably less complex than the original expression.