Hermine Smikle.
The Boolean algebra provides rigorous procedures for deciding whether a statement is true or false;if the statement can be expressed in two variables.
In Boolean algebra true is represented by a 1 and false by a 0. With these two digits (0,1) and the three basic operations called “not”, “and” and “or” digital algebra or switching algebra was developed.
The basic operations and their meaning:
Operations

Meanings


Symbols

or

Determine a single input

+ A+B


bit from the values of two or


more input.

and

Determines a single input

. A.B or AB


bit from the value of two


or more input

not

Changes binary bits to its

not A; bar over A


opposite value.


Any relationship between logical variables are called logical expressions. These expressions can be written as an equation for example the equation A + B + C = F where F is the name of the output variable. The expression A + B + C = F expresses the action of and/or function. Through Boolean Algebra logical analysis can be performed using these three functions.
The electronic representation of these functions are called logic gates. There are the and gate the not and the or gates. These logic gates are basic functional units for both arithmetic and logic operations; to operate they must accept binary numbers, and should have a carry bit of one or 0, (from the adjacent lower power of two), and should produce as outputs a sum bit and a carry bit for the next higher power of two.
How to design circuits
:
The first step in the design of a circuit is to establish a truth table that shows the output for all possible inputs.
Truth Tables
(a) A

A

(b)

A

B

A.B

(c) A

B

A + B


0

1


0

0

0


0

0


0


1

0


0

1

0


0

1


1


not



1

0

0


1

0


1





1

1

1


1

1


1






and




or

With proper input electronic digital circuits (logic circuits) establish logical manipulate paths. By passing binary signals through various combination of logic circuits, any desired information for computing can be operated on; each signal represents a binary carrying one “bit” of information.
The logic circuits or gates perform the logical operations.
Operations and their Gates:
Examples

1 And gate
(figure available in print form)
(figure available in print form)

____
Two and three input And gates.

2 Or gate
(figure available in print form)
(figure available in print form)

____
Two and three input Or gate.
3

Not (the not gate is sometimes called an inverter)

(figure available in print form)
____
not gate
Some Booleen functions have identical truth tables therefore their logic circuits serves identical purposes; but one may be preferable to the other. To do this more useful logic gates are created. The following gates NAND, and NOR were created for this purpose.
Examples of Truth Tables for Nand and Nor
Nand equation (figure available in print form).
A B F

Nand GATE

0 0 1

A

1 0 1

B

0 1 1
1 1 0
(figure available in print form)
Nor equation F = A + B

Nor gate

A B F
0 1 0
1 1 0
(figure available in print form)
Application To Digital Computer Circuits
Computer performs among other things, all kinds of arithmetic operations. The most basic operation is the addition of two binary digits, which consists of 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 and 1 + 0 = 1. The first three operations had sum that is a single digit, but 1 + 1 has a sum of two digits. the higher bit of this sum is called a carry. In adding two multiple digits numbers a carry is to be added to the next higher digit. The circuit that performs the addition of two bits is called a half adder. The circuit that performs the addition of three bits is called a full adder.
A half adder needs two binary inputs ( A and B) and two binary out puts ( S = sum; C= carry).
Truth Table for Half adder.
A B C S
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
The fourth row shows 1 + 1 = 10, here 1 is the carry to the next higher power of two.
S = A + B
C = AB
(figure available in print form)
In a full adder two of the input variable are shown by A and B, the third letter Z represents the carry from the previous lower position. The two outputs is denoted by S and C ( S = sum , C = carry)
Truth Table for Full Adder.
A B Z C S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0

(figure available in print form)

1 0 0 0 1

(figure available in print form)

1 1 0 1 0
1 1 1 1 1
Implementation of Boolean functions
: The translation of the Boolean function to logic circuits is called the implementation. The is the mathematical expression representing a combination of gates. Remember that a basic logic gate performs a single elementary logic operation and their inputoutput can be expressed as a logic expression.
These logic expression can be represented by a diagram.
GATES, LOGIC EXPRESSIONS AND THEIR DIAGRAMS
AND GATES
(figure available in print form)
(figure available in print form)
OR GATES
(figure available in print form)
(figure available in print form)
(figure available in print form)
(figure available in print form)
From the previous pages it can be seen that Boolean algebra facilitates the movement from a statement of the function, to truth table, then to a logical expression; these expressions can then be implemented in the form of a diagram using the numbers zero and one (1,0).
(figure available in print form)

Inputs

outputs


A B

F


0 0

0


0 1

1


1 0

1


1 1

1

OR GATE
The truth table explains the result F from the possible values of A and B
(figure available in print form)

Inputs

Outputs


A B C

F


0 0 0

0


1 0 0

1


0 1 1

1


1 1 1

1


0 0 1

1


1 1 0

1


1 0 1

1


0 1 1

1


1 1 1

1

If a small circle is placed at an input or output of a symbol for a logic gate it indicates negation.

1. If the small circle is placed at an input terminal, if the symbol entering is one the symbol leaving the circle and entering the block is 0.

2. If the circle is placed at the out block

____
(a) if the symbol leaving the block (and entering the circle) is 0, the symbol leaving the circle is 1.


____
Truth Table

Input

Output



A B C

F


1 0 0

1


0 0 0

1


1 1 0

1


1 0 1

1


0 1 0

1


0 0 1

1


1 1 1

0


0 1 1

1

(figure available in print form)
Boolean Algebra applied to Electrical problems
.
This idea can be applied to voltages that are present in a physical circuit. There are high voltages and low voltages. High voltage signifies that current is flowing, low voltage signifies that there is no current. These situations must be given some mathematical significance.
let high voltage

= 1

or

let high = 0

low voltage


= 0


low = 1


from these we can write truth tables to show the desired operations (or, and).
(figure available in print form)

Input

Output


A

B

F


low

low

low


low

high

high


high

low

high


high

high

high

or operation
Input

Output

input

output

A

B

F

A

B

F

0

0

0

1

1

1

0

1

1

1

0

0

1

0

1

0

1

0

1

1

1

0

0

0

In conclusion if the dualities of high and low, on and off are given values from the elements of the set of boolean algebra many physical electrical problems can be solved.