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Boolean algebra

Boolean algebra makes it possible to compare logical states. It is made up of three basic logical operators: AND, OR and NOT. Just with these 3 operators you can create any logical system.

1.1 The logical AND
Also called AND.
    Symbol :

AND

  

A and B are the inputs, out is the output.
Principle:

The relation between out, A and B is written: out = A. B:

   

  • if A = B = 0 then out = 0
  • if A = 1, B = 0 then out = 0
  • if A = 0, B = 1 then out = 0
  • if A = B = 1 then out = 1

In other words, the output is at 1 if and only if all the inputs are at 1.

We can establish this summary table (called truth table):

ABout
000
010
100
111

The AND can therefore be used to test if all the inputs of a system are at 1!

Note :
– we have studied an AND with 2 inputs, but we can put any number of them, the principle will remain the same: for the output to be at 1, all the inputs must be at 1.

1.2 The logical OR
Also called OR.

    Symbol :

OR

A and B are the inputs, out is the output.
Principle:

The relation between out, A and B is written: out = A + B:

  • if A = B = 0 then out = 0
  • if A = 1, B = 0 then out = 1
  • if A = 0, B = 1 then out = 1
  • if A = B = 1 then out = 1



In other words, the output is at 1 if at least one of the inputs is at 1.

Truth table:

ABout
000
011
101
111


With an OR, we can therefore see if at least one of the inputs of a system is at 1.

Note :
– you can put as many inputs as you want, the principle will remain the same: if at least one input is at 1, then the output is at 1.

1.3 The logical NOT

Also called INVERTER or NO.

    Symbol :

INVERTER

A is the input, S the output.

Principle:

The relation between Y and A is written: S = / A:

  • if A = 0 then S = 1
  • if A = 1 then S = 0



In other words, the output is the reverse of the input.

Truth table:

AS
01
10

1.4 Other operators …
   

There are other operators whose operation can be explained by the combination of the 3 basic operators.
For example, let’s take a look at the NAND operators:
NAND Symbol:


NAND

Principle:

   

We denote by S = /(A.B)
First of all, we can notice that this symbol is a “mix” between that of AND (for the half-oval) and that of NOT (for the small circle on the exit). This “mix” is also found in the name: NAND <=> NON + AND (ET).
For the operation, it is the same thing: it is about the mixture between an AND and a NO!
That is to say that this operator works in reverse AND: where there was a 1 at the output, there will be a 0; and vice versa.

We thus obtain the truth table of the NAND:

ABS
001
011
101
110

We therefore see that as soon as at least one input is at 0, then the output of the NAND is at 1.

Notes:

  • we can cite other operators:
  • the NOR = NOT OR. The NOR will function inversely to the OR.
  • the “exclusive OR”: it works like an OR, except in the case where all the inputs are at 1 because the output will be at 0.

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