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 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

Truth Table for 2x1 Karnaugh Map

Input	Output
A	X
0	X0
1	X1

Example 1.2

2x1 Karnaugh Map

	A
x0	x1

For the invertor example, this would have the values

Example 2.1

Truth Table for 2x1 Karnaugh Map

Input	Output
A	X
0	1
1	0

Example 2.2

2x1 Karnaugh Map

	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

Truth Table for 4x4 Karnaugh Map

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

4x4 Karnaugh Map

			A
				B
		x0	x1	x3	x2
C		x4	x5	x7	x6
	D	x12	x13	x15	x14
		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 ($\lnot$A$\land$$\lnot$B$\land$C$\land$D)$\lor$($\lnot$A$\land$B$\land$C$\land$D)$\lor$(A$\land$$\lnot$B$\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
	D	1	1	1	1
		0	0	0	0