Karnaugh Map (OR) K‒Map

KARNAUGH MAP (OR) K‒MAP

Simplifying Boolean functions using Boolean Algebra and theorems takes a lot of time. At every step, we need to rewrite the expression and apply different rules, which makes the process lengthy and difficult.

Boolean functions using Boolean postulates and theorems. It is a time consuming process and re-write the simplified expressions after each step.

To overcome this difficulty, Karnaugh introduced a method for simplification of Boolean functions in an easy way. This method is known as Karnaugh  map method or K-map method. It is a graphical method, which consists of 2n cells for 'n' variables. The adjacent cells are differed only in single bit position.

K-Map method is most suitable for minimizing Boolean functions of 2 variables to 5 variables.

All the cell order values of 2‒variable (Fig.2.16A) 3‒variable (Fig. 2.16B) and 4‒variable (Fig. 2.16C) are shown below.

2‒Variable K‒Map

The number of cells in 2 variable K-map is four (22) since the number of variables is two. The following figure shows 2 variable K-Map.

• There is only one possibility of grouping 4 adjacent min terms.

• The possible combinations of grouping 2 adjacent min terms are {(m0, m1), (m2, m3), (m0, m2) and (m1, m3)}.

3‒ Variable K‒Map

The number of cells in 3 variable K-map is eight (23), since the number of variables is three. The following figure shows 3 variable K-Map.

• There is only one possibility of grouping 8 adjacent min terms.

• The possible combinations of grouping 4 adjacent min terms are {(m₀, m₁, m₃, m₂), (m₄, m₅, m₇, m₆), (m₀, m₁, m₄, m₅), (m₁, m₃, m₅, m₇), (m₃, m₂, m₇, m₆) and (m₂, m₀, m₆, m₄)}.

• The possible combinations of grouping 2 adjacent min terms are {(m₀, m₁), (m₁, m₃), (m₃, m₂), (m₂, m₀), (m₄, m₅), (m₅, m₇), (m₇, m₆), (m₆, m₄), (m₀, m₄), (m₁, m₅), (m₃, m₇) and (m₂, m₆)}.

• If x=0, then 3 variable K-map becomes 2 variable K-map.

4‒Variable K‒Map 

The number of cells in 4 variable K-map is sixteen (2⁴), since the number of variables is four. The following figure shows 4 variable K-Map.

• There is only one possibility of grouping 16 adjacent min terms.

• Let R₁, R₂, R₃, and R₄, represents the min terms of first row, second row, third row and fourth row respectively. Similarly, C₁, C₂, C₃ and C₄ represents the min terms of first column, second column, third column and fourth column respectively. The possible combinations of grouping 8 adjacent min terms are {(R₁, R₂), (R₂, R₃), (R₃, R₄), (R₄, R₁), (C₁, C₂), (C₂, C₃), (C₃, C₄), (C₄, C₁)}. If w=0, then 4 variable K-map becomes 3 variable K-map.

5 Variable K-Map

The number of cells in 5 variable K-map is thirty-two (2⁵), since the number of variables is 5. The following figure shows 5 variable K-Map.

• There is only one possibility of grouping 32 adjacent min terms.

• There are two possibilities of grouping 16 adjacent min terms. i.e., grouping of min terms from m₀, to m₁₅, and m₁₆ to m₃₁.

• If v=0, then 5 variable K-map becomes 4 variable K-map. SocialSciences

Minimization of Boolean Functions using K-Maps

The combination of inputs for which the Boolean function is '1', then the Boolean function, which is in standard sum of products form after simplifying the K-map.

Similarly, the combination of inputs for which the Boolean function is '0', then the Boolean function, which is in standard product of sums form after simplifying the K-map.

Follow these rules for simplifying K-maps in order to get standard sum of products form.

• Select the respective K-map based on the number of variables present in the Boolean function.

• If the Boolean function is given as sum of min terms form, then place the ones at respective min term cells in the K-map. If the Boolean function is given as sum of products form, then place the ones in all possible cells of K-map for which the given product terms are valid. 

• Check for the possibilities of grouping maximum number of adjacent ones. It should be powers of two. Start from highest power of two and upto least power of two. Highest power is equal to the number of variables considered in K-map and least power is zero.

• Each grouping will give either a literal or one product term. It is known as prime implicant. The prime implicant is said to be essential prime implicant, if at least single '1' is not covered with any other groupings but only that grouping covers.

• Note down' all the prime implicants and essential prime implicants. The simplified Boolean function contains all essential prime implicants and only the required prime implicants.