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 :**

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):

A | B | out |

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

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 :

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:

A | B | out |

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

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 :

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:

A | S |

0 | 1 |

1 | 0 |

**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:

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:

A | B | S |

0 | 0 | 1 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

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.