5 Variable K Map
Online Karnaugh Map Calculator This online program generates the simplified function based on the input and output values of a function using Karnaugh Maps method. Enter the output values as minterm canonical form and the tool will calculate the simplified function.
The five-variable K-map is in effect two four-variable K-maps drawn horizontally to form an extension of each other. The two four-variable maps are designated as a = 0 and a = 1, respectively. Thus, the map a = 0 represents cells designated from 00000 (≡decimal 0) to 01111 (≡decimal 15) and the map a = 1 represents cells designated from 10000 (≡decimal 16) to 11111 (≡decimal 31). The simplification using the five-variable K-map is slightly complex and requires a lot of careful inspection of the two constituent maps. We shall now consider an example to illustrate the simplification process in a five-variable K-map. Solution: We know that adjacent-cell entries can be reduced by grouping them. 2.27, we have two separate charts or maps.
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How do we go about to find the adjacencies? There is no real difficulty in this if we consider the fact that the two charts themselves are adjacent to one another with the main element a = 0 for the first map and a = 1 for the second map. For example, the entries in the zeroth cell and the sixteenth cells are adjacent to each other because the zeroth cell represents 00000 and the sixteenth cell 10000. The two cells differ in the variable ‘ a’ only and this gets eliminated if we group the entries in cells 0 and 16 together.
In fact, a gets eliminated whenever the entries in corresponding cells of both the charts can be grouped together. Based on this principle, the entries of the given example are grouped, as shown in Fig. Now, consider Group 1 in the figure. We have a quad in chart a = 0 and a similar quad in chart a = 1.
Since these two quads occupy similar positions in the two charts, we group them together to form an octet. The resultant terms for Group 1 is b′c′. Group 2 has entries in chart 1 only, and reduces to a ′ bd′ e′. Group 3 has a single entry in chart 1, and a single entry in chart 2. The reduced term then is bcd ′ e′. Group 4 is similar to group 3 and reduces to bcde. Finally group 5 gives ab ′ de′.
5 Variable K Map
The solution, therefore, is.
I'm reaching back into my high school days trying to remember one of the rules about Karnaugh Maps. I have an 8 variable input, and I remember that I should try and make the selections a big as possible.
However, I vaguely remember that if I have a selection, that the number of $1$'s for a variable must equal the number of $0$'s for the same variable if it is to cancel out.When dealing with 4 variable tables, this is not a problem, but when I move to 8, I can get a single selection that has columns $0010$, $0110$, $0111$, $0101$, $0100$, $1100$, $1101$ and $1111$ selected.As you can see, the bit 3 (msb) has 5 $0$s and 3 $1$s, bit 2 has 1 $0$s and 7 $1$s, bit 1 and 0 have 4 of each. My questions are:.
Is this a valid single selection?. If this is a valid single selection, how do I handle it?. If not a valid single selection, why not? What are the rules that stipulate what a valid single selection is? What is the meaning behind those rules?I understand that when a bit is constant, it is used. When there are equal numbers of $1$s and $0$s, they cancel each other out.
So what is done when there unequal number of $1$s and $0$s? EDITFor those of you who didn't understand what I was talking about, here is a 8 variable Karnaugh map:The top header shows the numeric and the symbol representation of the values. I've not done the same for the left side header, but assume that they represent 4 other independent variables E, F, G, and H, though it is really irrelevenat.Assume that all that are marked $1$ are true values and those not marked are false.The $1$'s represent an 8x8 selection matrix, which by my understanding is a valid selection. Note that this could easily be an 8x1, 8x2, 8x4, 8x8 or 8x16 matrix located as a single grouping anywhere in the column specified.
This is just an example. Your 'Karnaugh Map' is not a valid Karnaugh map, because the ordering of the rows and columns does not follow the symmetry rules. Look at the example shown below.While Karnaugh maps are mainly used upto six variables, Mahoney extended the construction to more variables using reflection symmetries. An explains some details.
However, I have never seen this being used in practice.Your sample depicted as a map for 8 variables looks as follows:This example was created using a home-grown tool similar to.For 8 variables, there are 256 cells in the map. This is not really practical for human inspection.The trick of Karnaugh maps is to quickly find adjacent minterms which only differ in one input variable and can thus be merged into a term with fewer inputs.
However, as you can see from the example, this gets out of hand if the number of terms becomes too big.In a Mahoney map, '1' cells can be merged into a common block, if they can be folded onto each other in terms of a reflection symmetry.To minimize an expression with more than a handful of variables, tools like or are available for free. These are which strive to reduce a list of minterms.In case your original expression is a Boolean formula rather than a set of minterms, you might have a look at tools like, or.As a classic text on the topic, I can highly recommend:R. McMullen and A.