eight maxterms

Minterm vs Maxterm Solution

1. Identify the minterm (product term) term to be mapped.
2. Write the corresponding binary numeric value.
3. Use binary value as an address to place a 1 in the K-map.
4. Repeat steps for other minterms (P-terms within a Sum-Of-Products).

This logical sum is known commonly as Boolean addition as an OR function produces the summed term of two or more input variables, or constants. Thus the Boolean equation for a 2-input OR gate is given as: Q = A+B, that is Q equals both A OR B.

DeMorgan’s Theorems are basically two sets of rules or laws developed from the Boolean expressions for AND, OR and NOT using two input variables, A and B. These two rules or theorems allow the input variables to be negated and converted from one form of a Boolean function into an opposite form.

The sum-of-products (SOP) form is a method (or form) of simplifying the Boolean expressions of logic gates. In this SOP form of Boolean function representation, the variables are operated by AND (product) to form a product term and all these product terms are ORed (summed or added) together to get the final function./span>

Example 2: Minterm = AB’C’

1. First, we will write the minterm: Minterm = AB’C’
2. Now, we will write 0 in place of complement variables B’ and C’. Minterm = A00.
3. We will write 1 in place of non-complement variable A. Minterm = 100.
4. The binary number of the minterm AB’C’ is 100. The decimal point number of (100)2 is 4.

For SOP, we pair 1 and write the equation of pairing in SOP while that can be converted into POS by pairing 0 in it and writing the equation in POS form. For example, for SOP if we write x⋅y⋅z then for pos we write x+y+z.

maxterm (standard sum term) A sum (OR) of n Boolean variables, uncomplemented or complemented but not repeated, in a Boolean function of n variables. With n variables, 2 n different maxterms are possible. The complement of any maxterm is a minterm. See also standard product of sums. A Dictionary of Computing.

A Karnaugh map (K-map) is a pictorial method used to minimize Boolean expressions without having to use Boolean algebra theorems and equation manipulations. A K-map can be thought of as a special version of a truth table . Using a K-map, expressions with two to four variables are easily minimized.

Example. Karnaugh maps are used to facilitate the simplification of Boolean algebra functions. For example, consider the Boolean function described by the following truth table. are the maxterms to map (i.e., rows that have output 0 in the truth table).

The “Don’t care” condition says that we can use the blank cells of a K-map to make a group of the variables. To make a group of cells, we can use the “don’t care” cells as either 0 or 1, and if required, we can also ignore that cell.

Don’t-care values in a Karnaugh map relate to output that is generally not reachable under normal circumstances and allow you to simplify the logic more than you would normally be able to.

Boolean algebra is used to simplify Boolean expressions which represent combinational logic circuits. It reduces the original expression to an equivalent expression that has fewer terms which means that less logic gates are needed to implement the combinational logic circuit.

Don’t cares in a Karnaugh map, or truth table, may be either 1s or 0s, as long as we don’t care what the output is for an input condition we never expect to see. There is no requirement to group all or any of the don’t cares. Only use them in a group if it simplifies the logic.

The disadvantage of k map :

• It is not suitable for computer reduction.
• It is not suitable when the number of variables involved exceed four.
• Care must be taken to field in every cell with the relevant entry, such as a 0, 1 (or) don’t care terms.

Point to remember: While designing K-Map using SOP form, don’t care conditions (X) are considered as 1, if it helps form the largest group, otherwise it is considered as 0 and are left during encircling./span>

Pairs, Quads, and Octets Quad: A group of 4 one’s that are horizontally or vertically adjacent. End to end or in form of a square. A quad eliminates two variables and their complements. An octet eliminates three variables and their complements.

Redundant Group : It is a group whose all 1’s are overlapped by other groups. Redundant groups must be removed. Removal of redundant group leads to much simpler expression. Ex. 1 : Represent the following boolean expression in a K-map and simplify.

1. Groups may not include any cell containing a zero.
2. Groups may be horizontal or vertical, but not diagonal.
3. Groups must contain 1, 2, 4, 8, or in general 2n cells.
4. Each group should be as large as possible.
5. Each cell containing a one must be in at least one group.
6. Groups may overlap.
7. Groups may wrap around the table.

Steps to solve expression using K-map-

1. Select K-map according to the number of variables.
2. Identify minterms or maxterms as given in 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 maxterms(1’s elsewhere).

Simplify the given 2-variable Boolean equation by using K-map. We put 1 at the output terms given in equation. In this K-map, we can create 2 groups by following the rules for grouping, one is by combining (X’, Y) and (X’, Y’) terms and the other is by combining (X, Y’) and (X’, Y’) terms./span>