Karnaugh Maps

It may be necessary to read the section on Boolean Algebra in order to understand some of the notation used within this article.

A Karnaugh Map makes simplification of truth tables into functions much easier, but it is quite an involved concept to learn. You get all the minterms, and draw a rectangle around them using the fewest rectangles possible that include all the selected minterms. A rectangle must have 2^N minterms within it (so 1,2,4,8,16 etc.) The size of Karnaugh map depends on the number of variables dealt with.

In order to begin teaching the concepts behind Karnaugh maps, I will start with a 2x1 grid. This would not actually be used, but gives an example of how Karnaugh Maps work. I will use the example of an inverter. If a 1 is put in, a 0 is given out. If a 0 is put in, a 1 is given out. I will use the term xn to describe the letter x followed by a number. xn would normally be replaced by a 0, a 1 or a don't care term, represented by a - as is shown in examples 2.1 and 2.2 below. As you can see, everything below the A is the value for when A = 1. Everything which doesn't have an A above is the value for when A = 0. This may not make much sense yet, but should start to when we deal with larger maps.

Example 1.1
Input	Output
A	X
0	X0
1	X1

Example 1.2
	A
x0	x1

For the invertor example, this would have the values

Example 2.1
Input	Output
A	X
0	1
1	0

Example 2.2
	A
1	0

Below, I've shown how the output(X) of the truth table (normally 0s or 1s) are mapped onto a 4x4 karnaugh map.

Example 3.1
Inputs				Output
A	B	C	D	X
0	0	0	0	X0
0	0	0	1	X1
0	0	1	0	X2
0	0	1	1	X3
0	1	0	0	X4
0	1	0	1	X5
0	1	1	0	X6
0	1	1	1	X7
1	0	0	0	X8
1	0	0	1	X9
1	0	1	0	X10
1	0	1	1	X11
1	1	0	0	X12
1	1	0	1	X13
1	1	1	0	X14
1	1	1	1	X15

Example 3.2
			A
				B
		x0	x1	x3	x2
C		x4	x5	x7	x6
C	D	x12	x13	x15	x14
	D	x8	x9	x11	x10

In use, each occurrence of X and a number (Xn) would be replaced with a 0, 1 or - (don't care) term.

The letters at the top and sides show where each of the input variables (A,B,C and D) equal 1. For the Karnaugh map to be translated into a function, it is necessary to group similar terms. An example of this is shown below. The function ( $\neg$ A $\land$ $\neg$ B $\land$ C $\land$ D) $\lor$ ( $\neg$ A $\land$ B $\land$ C $\land$ D) $\lor$ (A $\land$ $\neg$ B $\land$ C $\land$ D) $\lor$ (A $\land$ B $\land$ C $\land$ D) is mapped below. It can be seen that are equal to 1 are shaded. All the shaded cells fall into both C and D. Therefore, the simplified function is C $\land$ D. Don't care terms can be assigned to either a 1 or a 0 depending on which leads to a more simplified solution.

Truth Table for 4x4 Karnaugh Map
Inputs				Output
A	B	C	D	X
0	0	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	1	1	0
0	1	0	0	0
0	1	0	1	0
0	1	1	0	0
0	1	1	1	0
1	0	0	0	0
1	0	0	1	0
1	0	1	0	0
1	0	1	1	0
1	1	0	0	1
1	1	0	1	1
1	1	1	0	1
1	1	1	1	1

4x4 Karnaugh Map
			A
				B
		0	0	0	0
C		0	0	0	0
C	D	1	1	1	1
	D	0	0	0	0

Example: 4-variable function with don’t-care conditions

Consider the function:

f(A, B, C, D) = Σm(3, 7, 11, 15) + d(1, 2, 5, 9, 10, 13, 14)

Here, d represents don't-care conditions. These correspond to input combinations that will never occur in practice, so they can be treated as either 0 or 1 in order to simplify the function.

The corresponding K-map is:

By grouping both 1s and don’t-cares, the simplified expression becomes:

f(A, B, C, D) = CD + AB

Example: Expression-to-K-map visualization

Some tools allow a Boolean expression to be entered directly and automatically generate the corresponding Karnaugh map.

For example, the expression:

f(A, B, C) = A'B'C+ABC'+BC

is mapped as shown below:

This approach eliminates the need to manually construct the truth table before drawing the map. It is especially useful for quickly verifying expressions and understanding how algebraic terms translate into grouped cells in the K-map.

Resources

Interactive Karnaugh Map Solver – TruthTableTools.com

 Free online solver supporting 2–4 variables, don’t-care conditions, and direct expression-to-K-map visualization.