### Boolean Algebra

In the following circuit, a bulb is controlled by two switches.

This control mechanism is denoted as **A.B** - **A and B** - in Boolean Algebra. The state of the switch is
The output is considered as **1**, when it is **on** and **0** when it is **off**. The way the bulb responds is considered as the
**output** and its state can also be described in terms of **o** -
**off** - and **1** - **on**.

On this basis, this animation can be summarized
in the table as follows:

A | B | A.B | Output |

1 | 0 | 0 | 0 |

0 | 1 | 0 | 0 |

0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 |

The following animation shows, the Boolean notation for the arrangement
of the switches is **A + B - A or B**.

The animation and the corresponding truth table explains how it works.

A | B | A + B | Output |

1 | 0 | 1 | 1 |

0 | 1 | 1 | 1 |

0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 |

The tables of this kind are called **Truth Tables**. A truth table describes how an arrangement of
switches controls the output of a logic circuit.

**E.g.**
The animations and corresponding truth tables show
how Boolean algebra is interpreted in real world.

A |
B |
B |
A |
Output |

1 |
0 |
1 |
0 |
1 |

0 |
1 |
0 |
1 |
1 |

0 |
0 |
1 |
1 |
1 |

1 |
1 |
0 |
0 |
0 |

B means the inverse of B; If B is **1**, B
is **0** and vice versa.

This is another truth table with its corresponding
circuit:

A |
B |
C |
A |
C |
Output |

1 |
1 |
1 |
0 |
0 |
1 |

1 |
0 |
0 |
0 |
1 |
0 |

1 |
0 |
1 |
0 |
0 |
0 |

1 |
1 |
0 |
0 |
1 |
1 |

0 |
0 |
0 |
1 |
1 |
0 |

0 |
1 |
0 |
1 |
1 |
0 |

1 |
0 |
1 |
0 |
1 |
1 |

0 |
1 |
1 |
1 |
0 |
1 |

Now try the following with a circuit and a
truth table:

- C.(A.B + A)
- A.C + A.B.C + A.B

- A.B.C.(A+B+C)
- A.(A.B.C + B.(A+C))
- C.(A+B).A