Defining Karnaugh Maps

A Karnaugh map provides a graphical representation of Boolean expressions in a tabular format, allowing for the extraction of simplified expressions by identifying patterns within the table. In a K-map, false is represented by zero, and true by one. To construct a K-map, follow these steps:

1. Identify the minterms related to your expression.
2. Generate Gray code sequences for the inputs.
3. Create and label your table.
4. Group the minterms in your table using rectangles.
5. Derive a simpler expression from the constants represented in those rectangles.

Now, let’s delve into each of these steps in detail.

Identifying Minterms

In Boolean algebra, a variable A can take the value of either zero or one. The complement of A, denoted as A' (A prime) or Ā (A bar), represents its opposite. For instance, if A equals one, then Ā equals zero, and vice versa. A minterm is defined as a product term that:

• Includes all inputs in either their original or complemented form.
• Evaluates to one.

For example, with inputs A, B, and C, the term ĀC cannot qualify as a minterm. However, ABC would be considered a minterm only if A, B, and C are all equal to one.

To determine the minterms of an expression, constructing a truth table is often the most straightforward approach. For instance, with the expression F = ĀB + AB, the truth table looks as follows, with the highlighted rows indicating the minterms.

The first video titled "Karnaugh Maps – Simplify Boolean Expressions" provides an excellent overview of how to utilize K-maps for Boolean simplification.

Generating Gray Code

Upon examining the truth table for F = ĀB + AB, a pattern emerges: each adjacent row changes by just one bit. This sequential arrangement is known as Gray code, sometimes referred to as reflected binary code. The Gray code sequence for a single variable is:

0

1

To create a sequence for two variables, reflect the column over a horizontal axis:

0

1

1

0

For three variables, the process continues similarly, resulting in a more complex arrangement:

0 0

0 1

1 1

1 0

1 0

1 1

0 1

0 0

Drawing Your K-map

For an expression with n inputs, a K-map will consist of 2ⁿ cells. For the expression F = ĀB + AB, which has two inputs, you would draw a table with four cells. I recommend labeling the rows and columns with the one-bit Gray code sequence to improve clarity.

Next, indicate the minterms on the K-map by marking the cells with ones.

Grouping Minterms

Adjacent ones in the K-map signify that simplification of the Boolean expression may be achievable. It's essential to draw rectangles around these groups, ensuring:

• The rectangles are as large as possible.
• The dimensions of the rectangles must be powers of two.

For F = ĀB + AB, a possible grouping could look like this:

You can also wrap rectangles around the table as needed.

Utilizing These Groups for Simplification

At this stage, the constants within the rectangles will guide you in formulating a new expression. For instance, if B remains constant while A varies, you can represent this group using the variable B. Therefore, the expression F = ĀB + AB simplifies to F = B.

To validate this, you can refer back to the truth table, confirming that the columns for F and B match.

In instances with multiple rectangles, you would express each rectangle in terms of its constants and then combine those results.

For example, in a different K-map, you might identify:

• An orange rectangle where B is constant at zero and C is one, resulting in B̅C.
• A blue rectangle with A and C both constant at one, yielding AC.

Thus, the overall simplified expression for that K-map would be F = B̅C + AC.