4 Variable Karnaugh Map III Video Lecture | Digital Electronics - Electrical Engineering (EE)

having introduced Carnot maps with two
and three variables in my previous video
let's take a look at some Carnot maps
with four variables here's a truth table
with four variables and a car no map to
match input values for a and B have been
written across the top notice that each
pair of bits differs from the previous
pair or the next pair by only one bit
similarly the input values of C and D
have been written down the side as row
labels and all of the possible
combinations of C and D have been
covered here finally we've placed the
output values of the truth table into
the Carnot map using the input values
ABC and D as coordinates so what does
this Carnot map tell us we can see we
have two groups of ones here the
vertical group always corresponds to a
value of one for input a so this group
represents a this group also always
corresponds to a value of one for input
B so this group represents a and B the
horizontal group on the other hand
always corresponds to a value of 0 for
input C so this group represents not see
these ones also correspond to an input
value of 1 for D so this group also
corresponds to D so we have not C and D
the two groups are related by the or
operator so here's our final expression
here's another example this time we've
got two groups of ones one of the groups
is using the table wrapping rule for our
first group we can see that where there
are one's the input value of a is always
zero so these represent not to pay
furthermore the value of C is always
zero so they represent not C
and finally whether a once the input
value of D is always one so this group
corresponds to D this group is therefore
not a and not C and D now let's take a
look at the other group where there are
ones the value of a is always one where
there are ones the value of D is always
zero so we have not D so the second
group corresponds to a and not D giving
us an overall expression as shown here
in this example we have a lot of ones we
may be tempted to make a group out of
these twelve ones but remember a group
can only contain one two four eight in
other words two to the power n ones so
this group is not valid we may be
tempted to say well there's a group of
eight but in order to do this we would
end up having to create three more
groups giving us a total of four groups
one of our rules states that there must
be as few groups as possible but if we
look more carefully we can see we've got
a group of eight ones here by applying
the table wrapping rule this means we
just need two more groups of eight which
overlap each other as shown here now we
have a total of three groups let's take
a look at these groups one at a time
this group represents not D you can see
that the value of input D is always zero
when there's a one but this group is
independent of the values of all of the
other inputs this group on the other
hand is always the opposite of a so this
group represents not a it's independent
of all of the other inputs and in this
final group you can see that all of the
ones correspond to an input value for B
one so this group represents B now
because these are three separate groups
we relate them using the or operator so
our final expression is not d or not a
or b you can see then for karnaugh maps
with four variables the same rules apply
as do for Carnot maps with two and three
variables