Karnaugh Maps / K-maps
Before going through the Karnaugh maps, some terms need to be clarified. These are as follows—
A literal is a single logic variable or its complement. For example— X, Y, A’, Z, X’, etc.
A minterm is the product of all the literals with or without complement involved in a logic system.
AB, A’B, AB’, A’B’ (for a problem containing only A and B),
A’BC, ABC, AB’C… (for a problem containing A, B and C)
When the values of different variables are given, minterms can be easily formed as-
If X=0, Y=0 minterm would be X’Y’
If X=1, Y=0, Z=1 minterm would be XY’Z
So, use the variable with value 1 as it is and variable with value 0 as complemented to find the minterm.
A maxterm is the sum of all the literals with or without complement involved in a logic system.
A+B, A+B’, A’+B, A’+B’ (for problem containing only A and B)
A+B+C, A’+B+C’, … (for problem containing A, B and C)
When the values of different variables are given, maxterms can be easily formed as-
If X=0, Y=0 maxterm would be X+Y
If X=1, Y=0, Z=1 maxterm would be X’+Y+Z’
So use the variable with value 1 as complemented and variable with value 0 as it is to find the maxterm.
A Boolean expression containing entirely of minterms or maxterms is known as canonical expression. These are of two types—
It is the product of all the maxterms that result in a false value of the output variable. For example—
The minterms and the maxterms can be represented by shorthand notation which makes it very easy and fast to write and work with. Shorthand notations can be obtained as follows-
To represent a minterm as shorthand notation following steps are to be followed-
1. Write 0 for a complemented term and 1 for non-complemented term. This will give you a binary number.
2. The shorthand notation will be an ‘m’ with the decimal equivalent of the binary number as subscript of ‘m’.
The minterm XYZ’ is represented as- XYZ’ will be represented as— XYZ’ --> 110 => So we get m6
To represent a maxterm as shorthand notation following steps are to be followed-
Write 1 for a complemented term and 0 for non-complemented
The shorthand notation will be a capital ‘M’ with the decimal term. This will give you a binary number equivalent of the binary number as subscript of ‘M’.
Eg. The minterm X+Y’ is represented as- X+Y+Z’ will be represented as— X+Y’+Z --> 010 => So M2
Minterm expansion of an expression
Any expression can be represented using minterms. To find the minterm expansion of an expression following steps have to be followed-
1. Write down all the terms in the expression
2. Put X where ever a literal is missing to convert the terms to minterms
3. Use all the combinations of Xs to find minterms
4. Remove the duplicate/repeated terms and write the terms together.
Complete Tutorial with Examples :
Here's a list of all the tutorials we currently have in this area - Introductory Digital Electronic Circuits and Boolean logic