Question

Minimize the following minterm function containing 'don't cares' using the Karnaugh-Map. $$ \mathrm{f}(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D})=\sum \mathrm{m}(5,6,7,8,9)+\mathrm{d}(10,11,12,13,14,15) $$

Short answer

The minimized function for the given minterm function with "don't cares" using the Karnaugh-Map method is \(f(A, B, C, D) = AB + B'D\).

Step-by-step solution

Convert decimal numbers to binary

First, let's convert given decimal minterm numbers and "don't care" terms into 4-bit binary numbers. - Minterms (5,6,7,8,9) in binary: 0101, 0110, 0111, 1000, 1001 - Don't cares (10,11,12,13,14,15) in binary: 1010, 1011, 1100, 1101, 1110, 1111

Creating the Karnaugh-Map

Create a Karnaugh-Map with 4 variables (A, B, C and D), as shown below: ``` CD 00 01 11 10 AB 00 0 0 X X 01 1 1 X X 11 1 1 X X 10 1 1 X X ``` In the above map, 1 represents the minterms where the function is true, 0 represents the minterms where the function is false, and X represents the "don't care" terms. The 16 cells of the map correspond to all possible values of the 4 variables (A, B, C, and D).

Grouping Minterms

Now, we need to group the minterms (1s) and "don't cares" (Xs) together to find the largest possible group of adjacent cells. Keep in mind that groups must have a size of \(2^n\) (where n is an integer) and wrap around the edges if needed. In this case, we can create two groups of 4 cells: - Group 1: Four 1s in the lower left quadrant covering all of row 11 and row 10. - Group 2: Four cells covering the two 1s in row 01 and two Xs in row 00 (01 column and 11 column)

Simplify the Boolean Expression

For each group, eliminate the variable that changes its value within the group: - Group 1: A doesn't change (always 1), B doesn't change (always 1), C changes (0 or 1), D changes (0 or 1) => \(AB\) - Group 2: A changes (0 or 1), B doesn't change (always 0), C changes (0 or 1), D doesn't change (always 1) => \(B'D\) Now, combine the simplified expressions of each group with an OR operation: Reduced function: \(f(A,B,C,D) = AB + B'D\)

Final Simplified Function

The final simplified function after minimizing using the Karnaugh-Map method is: $$ f(A, B, C, D) = AB + B'D $$ This is the minimized function for the given minterm function with "don't cares".

Key concepts

Boolean Algebra

Boolean algebra is a branch of algebra that deals with variables that have two distinct values, typically represented as true (1) and false (0). The basic operations are AND (conjunction), OR (disjunction), and NOT (negation), which follow specific rules to combine and manipulate these binary variables.

Minterms and Maxterms

In Boolean algebra, minterms and maxterms represent specific forms of Boolean expressions. A minterm is a product (AND operation) of literals where each variable appears exactly once, either in its original or complemented form. Conversely, a maxterm is a sum (OR operation) of literals where each variable also appears once in either form. These terms are fundamental in simplifying Boolean expressions, as seen in Karnaugh Map minimization.

Don't Care Conditions

In logical design and optimization, 'don't care' conditions refer to input combinations that do not affect the output of a circuit. During the Karnaugh Map minimization process, these conditions are represented by Xs and can be used to our advantage. We can consider 'don't care' cells as either 1 or 0, whichever assists in forming larger groups of adjacent cells to simplify the Boolean function further.

Binary Number Conversion

Binary number conversion is key in digital systems. To simplify Boolean functions using a Karnaugh Map, you first need to convert given decimal numbers to binary. Each decimal number is represented as a binary equivalent to construct a Karnaugh Map. For instance, the decimal number 5 is 0101 in binary, which informs us of the specific cell in the Karnaugh Map to which it corresponds.

Boolean Expression Simplification

Boolean expression simplification reduces complex Boolean equations to their simplest form, often to minimize the number of gates in a digital circuit. This involves applying rules of Boolean algebra, or through tools like the Karnaugh Map, which visualizes minterms and 'don't care' conditions. Groups of 1s and Xs in the map directly translate into simplified product terms that can be combined to form the most concise expression of the function.