Free Courses : The Combinational Logic and Algebra Of Boole

Logical Variable

George Boole, mathematician, logician and philosopher was born just 2 November 1815 in Lincoln, Lincolnshire (England).
It is the founding father of modern logic . In 1854 he succeeded where Leibniz had failed : to combine into a single mathematical language and symbolism.
The goal : to translate ideas and concepts into equations apply them certain laws and translate the results in logical terms .
To do this, it creates a binary algebra accepting only two numeric values ​​: 0 and 1.



The theoretical work of Boole, find essential applications in fields as diverse as computer systems, electrical and telephone lines, automation ...

Fundamentals :

Many electronic devices , electromechanical, mechanical, electrical, pneumatic, etc ... can take just two states ON or OFF.

Example:

• off on
• open closed
• true False
• block conduction 

For these reasons, it is much more advantageous to use a mathematical system using only two numeric values ​​( eg O or 1) to study the conditions of operation of these devices .

This is the system BIT

The set of mathematical rules that can be used with variables that can take only two possible values ​​is :

"Algebra Of Boole"

Concept of binary variable :

Logic variable is a variable that can take two values ​​, which are usually marked 0 or 1.
This variable is called binary and notes as a letter in algebra.

Example: a  b  x

Physically, this variable can be one of the devices mentioned above which two statements represent the two possible values ​​that can take this variable.

In general , these two statements are labeled H and L , and is assigned
the condition H (high) value of 1
in state L (low) value of 0

This notation zero is sometimes found : Ø to avoid confusion with the letter O.
The binary variable is also called Boolean variable.

Notion of logical function :

A logic function is the result of combination ( combinatorial logic ) of one or more interconnected logic variables by mathematical operations defined BOOLEAN :

the resulting value of this function depends on the value of the logical variables, but anyway this result can only be O or 1.
A logical function or has a variable input and a logic output variable logic .
This logic function is denoted by a letter as algebra.

Example:

• A G Y
• B F X

Notion of combinatorial logic

The combinational logic using logic functions , allows the construction of a combinational system .
A system is combinatorial when it is open loop type , ie no output is looped as input.
Each combination corresponds to a single input output . Combinatorial systems are simpler and can be represented by a truth table showing for each input state what is the status of the corresponding output .

Exercise:
Match by arrows, the terms left with the terms on the right:

Logic functions :

That a logical variable can take only two values ​​(0 or 1) , the number of functions is thereby limited.

1 variable logic function

Represent this variable with a reversing switch called "a"

Position L, we assign the value 0
Position H, we assign the value 1





For each of the following diagrams give the LED state V we take :
V = 1 if the light is on
V = 0 if the light is off

This gives us the following summary table :

There are no other possible combinations 

2 logical variables function

Let a and b logical variables can be represented by two independent switch reversing .

Considering firstly the two switches together , the
Four possible combinations of switching are :

Replace :
Position L by  0
Position H  by the value 1.

We get the following table:

Let us now examine the different possible functions that we can get from these 2 variables .
Different ways to connect these two switches to turn on an LED V lead to the following table:

( Value 0 if off)
( Value 1 if on)

Note : all 16 functions ( --- V0 > V15 ) has a value which depends on the chosen from 4 combination of variables b .

Let's Comment the following different functions:

V0 : The LED is always OFF = 0   }          Whatever the position of
V15 : The LED is always ON = 1  }                switches a and b

The LED V1 is ON if a and b are in position : 1
The LED V8 is OFF if a and b are in position : 0

The LED V3 is ON if b is in position : 1
independent of the position of a
The LED  V5 is ON if a is in position : 1
independent of the position of b

The LED V7 is ON if

• a is in position : 1
• or b is in position : 1
• or (a and b ) are in position 1

The LED V9 is ON if

• a and b are in position : 1
• or a and b are in position : 0
• but not if a ≠ b together

The LED V6 is ON if

• a is in position : 1
• or b is in position : 1
• but NOT if a = b simultaneously

Logical function of n variables :

In considering the two previous cases , we obtain:

For 1 variable   --->    2 combinations  --->  4 functions
For 2 variables  --->   4 combinations   ---> 16 functions
So, for n variable ---> 2ⁿ combinations --- > 2⁽²ⁿ⁾ functions

Examples:

3 variables ---> 8 combinations ---> 256 functions
4 variables ---> 16 combinations ---> 65,536 functions