Each individual cell is uniquely identified by the three Boolean Variables ( A, B, C). This 3-variable K-Map (Karnaugh map) has 2 3 = 8 cells, the small squares within the map. The regions are less obvious without color printing, more obvious when compared to the other three figures. In the final figure, we superimpose all three variables, attempting to clearly label the various regions. C occupies a square region in the middle of the rectangle, with C’ split into two vertical rectangles on each side of the C square. The variables B’ and B divide the universe into two square regions. The universe (inside the black rectangle) is split into two narrow narrow rectangular regions for A’ and A. We develop a 3-variable Karnaugh map above, starting with Venn diagram like regions. We do not necessarily enclose the A and B regions as at above left. Then, shade or enclose the region corresponding to B. Shade or circle the region corresponding to A. Mark the cell corresponding to the Boolean expression AB in the Karnaugh map above with a 1 The B below the diagonal is associated with the rows: 0 for B’, and 1 for B The 0 is a substitute for A’, and the 1 substitutes for A. The A above the diagonal indicates that the variable A (and A’) is assigned to the columns. The names of the variables are listed next to the diagonal line. The Karnaugh map above right is an alternate form used in most texts. For the sake of simplicity, we do not delineate the various regions as clearly as with Venn diagrams. In a similar manner B is associated with the cells to the right of it. The row headed by B’ is associated with the cells to the right of it. The column of two cells under A’ is understood to be associated with A’, and the heading A is associated with the column of cells under it. We don’t waste time drawing a Karnaugh map like (c) above, sketching a simplified version as above left instead. The lower right cell in figure (c) corresponds to AB where A overlaps B. The reason we do this is so that we may observe that which may be common to two overlapping regions-say where A overlaps B. We will now superimpose the diagrams in Figures (a) and (b) to get the result at (c), just like we have been doing for Venn diagrams. Imagine that we have go through a process similar to figures (a-f) to get a “square Venn diagram” for B and B’ as we show in middle figure (b). We need multiple variables.įigure (a) above is the same as the previous Venn diagram showing A and A’ above except that the labels A and A’ are above the diagram instead of inside the respective regions. What we have so far resembles a 1-variable Karnaugh map, but is of little utility. Also, we do not use shading in Karnaugh maps. We expand circle A at (b) and (c), conform to the rectangular A’ universe at (d), and change A to a rectangle at (e). ![]() Starting with circle A in a rectangular A’ universe in figure (a) below, we morph a Venn diagram into almost a Karnaugh map.
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