Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Examples: Most circuits are of the multiple output variety.
• A half adder adds two bits producing a single bit sum plus a single bit carry:
S := (x"y) ! (¬(x!y)) = x#y and C out := x!y. A half adder has two AND,
one OR, and one NOT gates.
• A full adder computes the complete two bit sum and carry out:
S := (x#y)#c in , where C in is the incoming carry. The carry is quite
complicated: C out := (xCy) + (yCC in ) + (C in Cx). A full adder has two half
adders and an OR gate.
• Ripple adders, lookahead adders, and lookahead carry circuits use many
bits as input to implement integer adders.
Half adder
Full adder
213
Note: Minimizing the Boolean algebra function means a less complicated
circuit. Simpler circuits are cheaper to make, take up less space, and are usually
faster. Add in how many devices are made and there is potentially a lot of
money involved in saving even a small amount of circuitry.
There are two basic methods for simplifying Boolean algebra functions:
• Karnaugh maps (or K-maps) provide a graphical or table driven technique
that works up to about 6 variables before it becomes too complicated.
• The Quine-McCluskey algorithm works with any number of variables.
Going to Google and searching on Karnaugh map software leads to a number of
programs to do some of the work for you.
Definition: A literal of a Boolean variable is its value or its complement. A
minterm of Boolean variables x 1 , x 2 , …, x n is a Boolean product of the {x i
,x i
}.
Note: A minterm is just the product of n literals.
214