Page 61 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Digital Circuits and Logic Design

Notes
Figure 4.2: Minterms

A Boolean expression will more often than not consist of multiple minterms corresponding to
multiple cells in a Karnaugh map as shown above. The multiple minterms in this map are the
individual minterms which we examined in the previous Figure 4.2. The point we review for
reference is that the 1s come out of the K-map as a binary cell address which converts directly to
one or more product terms. By directly we mean that a 0 corresponds to a complemented variable,
and a 1 corresponds to a true variable. Example: 010 converts directly to A’BC’. There was no
reduction in this example. Though, we do have a Sum-Of-Products result from the minterms.
Referring to the Figure 4.2, Let’s summarize the procedure for writing the Sum-Of-Products
reduced Boolean equation from a K-map:
• Form largest groups of 1s possible covering all minterms. Groups must be a power of 2.
• Write binary numeric value for groups.
• Convert binary value to a product term.
• Repeat steps for other groups. Each group yields a p-terms within a Sum-Of-Products.

Nothing new so far, a formal procedure has been written down for dealing with minterms. This
serves as a pattern for dealing with maxterms.

Next we attack the Boolean function which is 0 for a single cell and 1s for all others.
Figure 4.3: Maxterm in the K-map

A maxterm is a Boolean expression resulting in a 0 for the output of a single cell expression, and
1s for all other cells in the Karnaugh map, or truth table. The illustration above left shows the
maxterm(A + B + C), a single sum term, as a single 0 in a map that is otherwise 1s. If a maxterm
has a single 0 and the remaining cells as 1s, it would appear to cover a maximum area of 1s.
There are some differences now that we are dealing with something new, maxterms. The maxterm
is a 0, not a 1 in the Karnaugh map. A maxterm is a sum term, (A + B + C) in our example, not a
product term.
It also looks strange that (A + B + C) is mapped into the cell 000. For the equation
Out=(A + B + C) = 0, all three variables (A, B, C) must individually be equal to 0. Only (0 + 0 +
0) = 0 will equal 0. Thus we place our sole 0 for minterm (A + B + C) in cell A, B, C = 000 in the
K-map, where the inputs are all 0. This is the only case which will give us a 0 for our maxterm.

56 LOVELY PROFESSIONAL UNIVERSITY