• Larger logic problems require a systematic
approach for solution. Modern integrated
circuit chips can use millions of logic
devices. The sheer magnitude of these
designs is a clear sign that a formal
approach to the design is needed.
• Given a Boolean function described by a
– determine the smallest sum of products
function that has the same truth table.
– determine the AND-OR-NOT circuit that
implements that smallest sum of products
• Different ways of expressing a Boolean
function can have widely varying levels of
• More complex circuits will require more gates
and inverters, so it's a reasonable goal to learn
how to devise circuits that are as simple as
• In this section we are going to look at how you
can represent circuits differently using
• Let's look at a simple Boolean function of three variables. We'll
describe this function with a truth table. The input variables are
X, Y and Z, and the function output is F.
Let's examine this function in some detail. The only non-zero entries
X = 0, Y = 1, Z = 0
and X = 1, Y = 0, Z = 1
The function is 1 for those two input conditions and zero for all other
How we can implement this function
• We want the output to be 1 whenever we
– X=0 AND Y=1 AND Z=0
• OR when we have
– X=1 AND Y=0 AND Z=1.
• This word statement is very close to the
function we want. We've highlighted the
important aspects of the function. Here's the
• This function is read as (NOT-X AND Y AND
• This form is composed of two groups of three.
Each group of three is a minterm. Important
points about minterms include the following.
– In a minterm, each variable, X, Y or Z appears
once, either as the variable itself or as the inverse.
– Each minterm corresponds to exactly one entry
(row!) in the truth table.
• To build any Boolean function from
minterms do the following.
– Get a truth table for the function
– For each entry of the truth table for which
the function takes on a value of 1, determine
the corresponding minterm expression
remembering that every variable of its
inverse will appear in every minterm.
– OR all the minterms from the second step
IN SHORT !
• A truth table gives a unique sum-of-products
function that follows directly from expanding
the ones in the truth table as minterms.
An example using Minterms
• Three young graduates have formed a
company. The three
graduates, Alisha, Ben and Corey have a
system to minimize friction. For all minor
decisions they want to use a circuit that
will determine when a majority of the
three of them has voted for a proposal.
Essentially, they want a box with three
inputs that will produce a 1 at the output
If you OR anything with itself, you get the original
quantity back (X + X = X).
• Here is the three-variable truth table and
the corresponding minterms:
• Example: suppose a function F is defined by
the following truth table
Since F= 1 on rows 1, 2, 4, and 7, we obtain
A compact notation is to write only the
numbers of the minterms included in , using
the Greek letter capital sigma to indicate a