Prerequisite – Implicant in K-Map

**Karnaugh Map** or K-Map is an alternative way to write truth table and is used for the simplification of Boolean Expressions. So far we are familiar with 3 variable K-Map & 4 variable K-Map. Now, let us discuss the 5-variable K-Map in detail.

Any Boolean Expression or Function comprising of 5 variables can be solved using the 5 variable K-Map. Such a 5 variable K-Map must contain ** = 32 cells** . Let the 5-variable Boolean function be represented as :

** f ( P Q R S T) ** where P, Q, R, S, T are the variables and P is the most significant bit variable and T is the least significant bit variable.

The structure of such a K-Map for SOP expression is given below :

The cell no. written corresponding to each cell can be understood from the example described here:

Here for variable **P=0, we have Q = 0, R = 1, S = 1, T = 1 i.e. (PQRST)=(00111)** . In decimal form, this is equivalent to **7**. So, for the cell shown above the corresponding cell no. = 7. In a similar manner, we can write cell numbers corresponding to every cell as shown in the above figure.

Now let us discuss how to use a 5 variable K-Map to minimize a Boolean Function.

**Rules to be followed : **

- If a function is given in compact canonical SOP(Sum of Products) form then we write
**“1”**corresponding to each minterm ( provided in the question ) in the corresponding cell numbers. For eg:

For we will write “1” corresponding to cell numbers (0, 1, 5, 7, 30 and 31). - If a function is given in compact canonical POS(Product of Sums) form then we write
**“0”**corresponding to each maxterm ( provided in the question ) in the corresponding cell numbers. For eg:

For we will write “0” corresponding to cell numbers (0, 1, 5, 7, 30 and 31).

**Steps to be followed : **

- Make the largest possible size subcube covering all the marked 1’s in case of SOP or all marked 0’s in case of POS in the K-Map. It is important to note that each subcube can only contain terms in powers of 2 . Also a subcube of cells is possible if and only if in that subcube for every cell we satisfy that “m” number of cells are
*adjacent cells*. - All
*Essential Prime Implicants (EPIs)*must be present in the minimal expressions.

**I. Solving SOP function –**

For clear understanding, let us solve the example of SOP function minimization of 5 Variable K-Map using the following expression :

In the above K-Map we have 4 subcubes:

The one marked in red comprises of cells ( 0, 4, 8, 12, 16, 20, 24, 28)__Subcube 1:__The one marked in blue comprises of cells (7, 23)__Subcube 2:__The one marked in pink comprises of cells ( 0, 2, 8, 10, 16, 18, 24, 26)__Subcube 3:__The one marked in yellow comprises of cells (24, 25, 26, 27)__Subcube 4:__

Now, while writing the minimal expression of each of the subcubes we will search for the literal that is common to all the cells present in that subcube.

__Subcube 1__:__Subcube 2__:__Subcube 3__:__Subcube 4__:

Finally the minimal expression of the given boolean Function can be expressed as follows :

**II. Solving POS function –**

Now, let us solve the example of POS function minimization of 5 Variable K-Map using the following expression :

In the above K-Map we have 4 subcubes:

The one marked in red comprises of cells ( 0, 4, 8, 12, 16, 20, 24, 28)__Subcube 1:__The one marked in blue comprises of cells (7, 23)__Subcube 2:__The one marked in pink comprises of cells ( 0, 2, 8, 10, 16, 18, 24, 26)__Subcube 3:__The one marked in yellow comprises of cells (24, 25, 26, 27)__Subcube 4:__

Now, while writing the minimal expression of each of the subcubes we will search for the literal that is common to all the cells present in that subcube.

__Subcube 1__:__Subcube 2__:__Subcube 3__:__Subcube 4__:

Finally the minimal expression of the given boolean Function can be expressed as follows :

__NOTE__:

- For the 5 variable K-Map, the Range of the cell numbers will be from 0 to -1 i.e. 0 to 31.
- The above mentioned term
*“Adjacent Cells”*means that “any two cells which differ in only 1 variable”.

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