Minterm (All of the variables are AND'd together.)

An Example Function

The input variables are X, Y and Z, and the function output is F.  

x y z f0 0 0 00 0 1 00 1 0 10 1 1 01 0 0 01 0 1 11 1 0 01 1 1 0

non-zero entries are at:

 X = 0, Y = 1, Z = 0

& X = 1, Y = 0,  Z = 1

        Description of what we want to implement:

We want the output to be 1 whenever we have eithero X=0 AND Y=1 AND Z=0

OR when we haveo X=1 AND Y=0 AND Z=1.

Here's the function:

This kind of function is often referred to as a "sum of products" form.

Sum Of Products (SOP)    

The Sum of Products form represents an expression as a sum of minterms.

F(X, Y, ...) = Sum (ak.mk)

where ak is 0 or 1 and mk is a minterm. To derive the Sum of Products form from a truth table, OR together all of the minterms which give a value of 1.

Consider the truth table  

x y F Minterm0 0 0 X'.Y'0 1 0 X'Y1 0 1 X.Y'1 1 1 X.Y

Here SOP is f(X.Y) = X.Y' + X.Y

Note that m must always be lowercase. An uppercase 'M' represents a maxterm.

Maxterm (All variables are OR'd together.)

Product Of Sum (POS)

The Product of Sums form represents an expression as a product of maxterms.F(X, Y, .......) = Product (bk + Mk), where bk is 0 or 1 and Mk is a maxterm.To derive the Product of Sums form from a truth table, AND together all of the maxterms which give a value of 0.

Example – POS

x y F Maxterm0 0 1 X+Y0 1 0 X+Y'1 0 1 X'+Y1 1 1 X'+Y'

Here POS is F(X,Y) = (X+Y')

Exercise

Give the expression represented by the following truth table in both Sum of Products and Product of Sums forms.

x

y

z

F(X,Y,X)

0 0 0 10 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0

Conversion between POS and SOP

Conversion between the two forms is done by application of DeMorgans Laws.

Simplification

As with any other form of algebra you have encountered, simplification of expressions can be performed with Boolean algebra.

Example

Show that X.Y.Z' + X'.Y.Z' + Y.Z = Y

X.Y.Z' + X'.Y.Z' + Y.Z = Y.Z' + Y.Z = Y

Example

Show that (X.Y' + Z).(X + Y).Z = X.Z + Y.Z

(X.Y' + Z).(X + Y).Z

= (X.Y' + Z.X + Y'.Z).Z

= X.Y'Z + Z.X + Y'.Z

= Z.(X.Y' + X + Y')

= Z.(X+Y')

• Understand what is mean by the minterm (sum of product) and maxterm (product of sum)

Half Adder

The half adder is an example of a simple, functional digital circuit built from two logic gates.  The half adder adds to one-bit binary numbers (AB).  The output is the sum of the two bits (S) and the carry (C).

Note how the same two inputs are directed to two different gates.  The inputs to the XOR gate are also the inputs to the AND gate.  The input "wires" to the XOR gate are tied to the input wires of the AND gate; thus, when voltage is applied to the A input of the XOR gate, the A input to the AND gate receives the same voltage.

AND for carry,

XOR for sum.

Truth Table

Full adder

you see that you add three bits. Half adders only add two bits.

We need a circuit that can add three bits. That circuit is called a full adder.

Here are the characteristics of a full adder.

Data inputs: 3 (call them x, y, and cin, for carry in) Outputs: 2 (call them s, for sum, and cout, for carry out)

Notice we now need to make a distinction whether the carry is an input (cin) or an output (cout). Carry in's in column i are due to carry outs from column i - 1 (assuming we number columns right to left, starting at column 0 at the least significant bit).

Here's a truth table for full adders. Row x y cin cout s

0 0 0 0 0 0

1 0 0 1 0 1

2 0 1 0 0 1

3 0 1 1 1 0

4 1 0 0 0 1

5 1 0 1 1 0

6 1 1 0 1 0

7 1 1 1 1 1

Note:

You find above mention topics in detail I have just summary of them and some of the topics is missing truth table and circuit.

www.facstaff.bucknell.edu/mastascu/eLessonsHtml/Logic/Logic2.html

everything2.com/title/minterm

www.asic-world.com/digital/boolean2.html

www.cs.umd.edu/class/sum2003/cmsc311/Notes/Comb/adder.html

isweb.redwoods.cc.ca.us/INSTRUCT/CalderwoodD/diglogic/half-add.htm