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Introduction to Karnaugh Map (K-map)

A Karnaugh Map (K-map) is a graphical method to simplify Boolean algebra expressions. It is useful for minimizing expressions with up to 4 variables. K-maps provide visual map to see and eliminate redundancies in Boolean expressions, making use of efficient pattern recognition by human eye.

Why Use K-maps?

Boolean algebra can be time-consuming and error-prone, especially when dealing with more than 2 variables. K-maps offer a more intuitive and systematic approach to simplification. Here are some advantages:

• Visual Simplicity: K-maps provide a clear visual representation of the logic.

• Error Reduction: It reduces the chances of missing terms or making mistakes.

• Efficiency: Simplification is faster compared to algebraic methods.

Structure of a K-map

A K-map is a grid where each cell represents a unique combination of input variables. The number of cells depends on the number of variables:

• 2 variables: 4 cells (2x2 grid)

• 3 variables: 8 cells (2x4 grid)

• 4 variables: 16 cells (4x4 grid)

The cells are arranged so that adjacent cells differ by only one variable. This is called Gray code ordering.

Steps to Simplify Using K-maps

1. Create the K-map: Draw the grid based on the number of variables.

2. Fill the K-map: Place 1s in the cells corresponding to the minterms of the Boolean expression.

3. Group the 1s: Identify groups of adjacent 1s (groups should be powers of 2: 1, 2, 4, 8, etc.).

4. Write the simplified expression: For each group, identify the common variables and write the simplified term.

Example 1: Simplifying a 2-Variable Boolean Expression

Let’s simplify the Boolean expression: F(A, B) = Σ(0, 1, 2).

Step 1: Create a 2x2 K-map.

B\A| 0 | 1 |

---+---+---+

 0 | 1 | 1 |

 1 | 1 | 0 |

Step 2: Fill the K-map with 1s for minterms 0, 1, and 2.

Step 3: Group the 1s. Here, we can group the top row (minterms 0 and 1) and the first column (minterms 0 and 2).

Step 4: Write the simplified expression. The top row corresponds to A', and the first column corresponds to B'. So, the simplified expression is: F(A, B) = A' + B'.

Example 2: Simplifying a 3-Variable Boolean Expression

Let’s simplify the Boolean expression: F(A, B, C) = Σ(0, 2, 4, 6).

Step 1: Create a 2x4 K-map.

A\BC|00|01|11|10|

----+--+--+--+--+

0   |1 |0 |0 |1 |

1   |1 |0 |0 |1 |

Step 2: Fill the K-map with 1s for minterms 0, 2, 4, and 6.

Step 3: Group the 1s. Here, we can group all four corners (minterms 0, 2, 4, 6).

Step 4: Write the simplified expression. The group corresponds to C'. So, the simplified expression is: F(A, B, C) = C'.

Why K-maps Are Better Than Boolean Algebra

While Boolean algebra is powerful, it can become cumbersome for larger expressions. K-maps offer several advantages:

• Visual Clarity: K-maps provide a clear picture of the logic, making it easier to spot redundancies.

• Faster Simplification: Grouping 1s in a K-map is quicker than applying Boolean laws repeatedly.

• Error Detection: Mistakes are easier to spot in a K-map than in a long algebraic expression.

Karnaugh Maps - Rules of Simplification

The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing ones

1. Group can not include any cell containing a zero.

B\A| 0 | 1 |  B\A| 0 | 1 |

---+---+---+  ---+---+---+

 0 | 1 | 1 |   0 | 1 | 1 |

 1 | 1 | 0 |   1 | 1 | 0 |

    WRONG          RIGHT  

2. Groups can be horizontal or vertical, but not diagonal.

B\A| 0 | 1 |  B\A| 0 | 1 |

---+---+---+  ---+---+---+

 0 | 1 | 1 |   0 | 1 | 1 |

 1 | 1 | 0 |   1 | 1 | 0 |

    WRONG          RIGHT  

3. Groups size must be 1, 2, 4, 8, or in general 2n cells. Group size can not be 3, 5, 6 , 7 etc.

A\BC|00|01|11|10|   A\BC|00|01|11|10|   

----+--+--+--+--+   ----+--+--+--+--+  

0   |0 |1 |1 |1 |   0   |0 |1 |1 |1 |  

1   |0 |0 |0 |0 |   1   |0 |0 |0 |0 |  

     WRONG               RIGHT

4. Each group should be as large as possible.

A\BC|00|01|11|10|   A\BC|00|01|11|10|   

----+--+--+--+--+   ----+--+--+--+--+  

0   |0 |1 |1 |0 |   0   |0 |1 |1 |0 |  

1   |0 |1 |1 |0 |   1   |0 |1 |1 |0 |  

     WRONG              RIGHT

5. A cell containing a one must be in at least one group. 

A\BC|00|01|11|10|   A\BC|00|01|11|10|   

----+--+--+--+--+   ----+--+--+--+--+  

0   |1 |0 |1 |0 |   0   |1 |0 |1 |0 |  

1   |0 |0 |1 |0 |   1   |0 |0 |1 |0 |  

     WRONG              RIGHT

6. Groups may overlap. A cell containing 1 can be part of multiple groups.

A\BC|00|01|11|10|   A\BC|00|01|11|10|   

----+--+--+--+--+   ----+--+--+--+--+  

0   |1 |1 |1 |1 |   0   |1 |1 |1 |1 |  

1   |0 |1 |1 |0 |   1   |0 |1 |1 |0 |  

     WRONG              RIGHT

7. Groups will wrap around the table. The leftmost cell in a row can be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.

A\BC|00|01|11|10|   A\BC|00|01|11|10|   

----+--+--+--+--+   ----+--+--+--+--+  

0   |1 |0 |0 |1 |   0   |1 |0 |0 |1 |  

1   |1 |0 |0 |1 |   1   |1 |0 |0 |1 |  

     WRONG              RIGHT

8. The total number of groups should be as small as possible.

A\BC|00|01|11|10|   A\BC|00|01|11|10|   

----+--+--+--+--+   ----+--+--+--+--+  

0   |0 |0 |1 |1 |   0   |0 |0 |1 |1 |  

1   |0 |0 |1 |1 |   1   |0 |0 |1 |1 |  

     WRONG              RIGHT

Summmary:

1. No zeros allowed.

2. No diagonals.

3. Only power of 2 number of cells in each group.

4. Groups should be as large as possible.

5. No 1 should b left out without a group.

6. Overlapping allowed.

7. Wrap around allowed.

8. Fewest number of groups desired.

Boolean Algebra vs. Karnaugh Map

Let's simplify the following 3-variable Boolean expression using both Boolean algebra and Karnaugh Map (K-map). This example demonstrates the superiority of K-maps in simplifying complex expressions quickly and visually.

Example:

F(A, B, C) = Σ(1, 2, 3, 5, 7)

Simplification Using Boolean Algebra

Step 1: Write the expression in canonical SOP (Sum of Products) form:

F(A, B, C) = A'B'C + A'BC' + A'BC + AB'C + ABC

Step 2: Group terms to apply Boolean algebra rules:

F(A, B, C) = A'B'C + A'BC' + A'BC + AB'C + ABC

           = A'B'C + A'BC' + A'BC + AB'C + ABC

           = A'C(B' + B) + A'BC' + AB'C + ABC

Step 3: Simplify using Boolean identities (B' + B = 1):

F(A, B, C) = A'C(1) + A'BC' + AB'C + ABC

           = A'C + A'BC' + AB'C + ABC

Step 4: Factor common terms:

F(A, B, C) = A'C + A'BC' + C(AB' + AB)

           = A'C + A'BC' + C(A(B' + B))

Step 5: Simplify further (B' + B = 1):

F(A, B, C) = A'C + A'BC' + C(A(1))

           = A'C + A'BC' + AC

Step 6: Factor C:

F(A, B, C) = C(A' + A) + A'BC'

           = C(1) + A'BC'

           = C + A'BC'

Final Simplified Expression:

F(A, B, C) = C + A'BC'

Simplification Using Karnaugh Map (K-map)

Step 1: Draw the 3-variable K-map and fill in the minterms:

A\BC|  00  01  11  10

---------------------

0   |  0   1   1   1

1   |  0   1   1   0

Step 2: Group adjacent 1s:

A\BC|00 01 11 10   A\BC|00 01 11 10

----------------   ----------------

0   |0  1  1  1    0   |0  1  1  1

1   |0  1  1  0    1   |0  1  1  0

     Group-1             Group-2

We have two groups:

• Group 1: Covers minterms 1, 3, 5, 7 (Square group).

• Group 2: Covers minterms 2, 3 (Horizontal group).

Step 3: Simplify each group:

• Group 1: All cells in this group have C = 1, so the term is C.

• Group 2: All cells in this group have A = 0 and B = 1, so the term is A'B.

Final Simplified Expression:

F(A, B, C) = C + A'B

Comparison of Results

• Boolean Algebra: Simplified expression is F(A, B, C) = C + A'BC'.

• Karnaugh Map: Simplified expression is F(A, B, C) = C + A'B.

The K-map simplification is more efficient and avoids the redundant term A'BC' present in the Boolean algebra result. This demonstrates the superiority of K-maps in simplifying Boolean expressions quickly and accurately.