WHAT IS DIGITAL LOGIC?

• Modern computing system consists of complex system and technologies. These technologies are built upon some fundamental simple logics known as Digital Logic.
• By using digital logic gates we can develop complex logical circuit for various purposes like data storing, data manipulation or simply data representation.
• Digital logic is mainly used for data(must be digital information) representation, manipulation and processing of using discrete signals or binary digits (bits). It can perform logical operations, data retrieval or storing and data transformation by analyzing logical circuit design.
• Digital logic is also known as ‘Boolean logic’. The inputs and output can be termed as ‘Boolean Variables’. The output boolean variable of a digital signal can be expressed in terms of input boolean variables which forms the ‘Boolean Expression’.
• Representation of Boolean expression can be primarily done in two ways. They are as follows:
  1. Sum of Products (SOP) form
  2. Product of Sums (POS) form

SOP (SUM OF PRODUCTS) FORM

• The SOP stands for Sum of Products defined as the sum of minterms. As the name suggests the SOP form represents the OR operation of the product terms i.e., minterms.
• The SOP represents the minterms in the Boolean algebra. The SOP is denoted by ∑. As the SOP deals with the minterms it works on active high logic.

• Example of SOP:

  • AB + CD
  • P'Q + R
  • X'Y' + W'Z'

POS (PRODUCT OF SUM) FORM

• The POS stands for Product of Sum defined as the product of maxterms. As the name suggests the POS form represents the AND operation of the sum terms i.e., maxterms.
• The POS represents the maxterms in Boolean algebra. The POS is denoted by Π. As the POS deals with the maxterms it works on active low logic.

• Example of POS:

  • (A + B).(C + D)
  • (P' + Q).(R + S')

KARNAUGH MAP (K-MAP) METHOD

• The K-map is a systematic way of simplifying Booleann expressions. With the help of the K-map method, we can find
  the simplest POS and SOP expression, which is known as the minimum expression.
• Just like the truth table, a K-map contains all the possible values of input variables and their corresponding output
  values.
• In K-map, the number of cells is similar to the total number of variable input combinations. For example,
  if the number of variables is three, the number of cells is 2^3=8.

STEPS TO SOLVE K-MAP:

• Steps to Solve Expression using K-map:

  1. Select the K-map according to the number of variables.
  2. Identify minterms or maxterms as given in the problem.
  3. For SOP put 1’s in blocks of K-map respective to the minterms (0’s elsewhere).
  4. For POS put 0’s in blocks of K-map respective to the max terms (1’s elsewhere).
  5. Make rectangular groups containing total terms in power of two like 2,4,8 ..(except 1) and try to cover as many elements as you can in one group.
  6. From the groups made in step 5 find the product terms and sum them up for SOP form.

K-MAP FOR 3 VARIABLES:

SOP FORM:

• We can use the K-Map to simplify a Boolean function of three-variables. A Boolean function in three variables (A, B, C) can be expressed in the standard sum of product (SOP) form that can have total eight possible combinations, which are as follows −

               (A′B′C′),(A′B′C),(A′BC′),(A′BC),(AB′C′),(AB′C),(ABC′),(ABC)

• In terms of POS (Product of Sum) form, the eight possible combinations of the three variables Boolean expression are as follows-

           (A+B+C),(A+B+C′),(A+B′+C),(A+B′+C′),(A′+B+C),(A′+B+C′),(A′+B′+C),(A′+B′+C′)

• Summing these product terms  we get- Final expression = (A’C+AB) .

POS FORM:

• Final expression of the K-Map will be:

  • (A' + B’) (B’ + C’) (A + B + C) 

DON'T CARE CONDITION

• The ‘Don’t Care’ conditions allow us to treat certain cells in a K-Map as either 0, 1, or to ignore them altogether, which can help in forming larger and more efficient groups compared to grouping without ‘Don’t Care’ cells. 
• When forming groups, we can consider a ‘Don’t Care’ cell as either 0 or 1, or we can choose to ignore it. Therefore, the “Don’t Care” condition can help us to form a larger group of cells. 
• A standard Sum of Products (SOP) function with ‘Don’t Care’ conditions can be converted into a Product of Sums (POS) expression by retaining the ‘Don’t Care’ cells and converting the missing minterms of the SOP form into maxterms of the POS form. Similarly, a POS function having don’t cares can be converted to SOP form keeping the don’t cares as they are and writing the missing maxterms of the POS expression as the minterms of SOP expression. 

STEPS TO USE DON'T CARE CONDITION:

• Identify Don’t Care Conditions: Find out which combinations of variables result in don’t care conditions. These can be portrayed directly in the question.
• Place Don’t Care Values on the K-Map: Put the don’t care conditions in the K-Map with the help of symbols like X, -, or Φ.Form
• Groups with Don’t Care Values: Use don’t-care cells to extend the number of 1s and thus, form bigger groups. These groups help to further minimize the expression.
• Simplify the Expression: The simplified Boolean expression should be written using the grouped terms. Only don’t care values should be used to form larger groups.

• Example: Minimize the following function in SOP minimal form using K-Maps: 

                     f = m(1, 5, 6, 11, 12, 13, 14) + d(4) 

• Therefore, SOP minimal is:

              f = BC' + BD' + A'C'D + AB'CD