## Understanding (a+b)(c+d) in Boolean Algebra

In Boolean algebra, the expression **(a + b)(c + d)** represents the **product of two sums**, where 'a', 'b', 'c', and 'd' are Boolean variables that can hold either a '0' (False) or '1' (True) value. This expression is a fundamental building block in simplifying and manipulating Boolean functions, used widely in digital logic design.

### The Logic Behind the Expression

The expression **(a + b)(c + d)** can be interpreted as follows:

**(a + b)**: This represents the**sum**or**OR**operation between variables 'a' and 'b'. The result is '1' (True) if either 'a' or 'b' is '1', and '0' (False) otherwise.**(c + d)**: Similarly, this represents the**sum**or**OR**operation between variables 'c' and 'd'.**(a + b)(c + d)**: This represents the**product**or**AND**operation between the two sums. The result is '1' (True) only if both sums are '1' (True).

### Expanding the Expression

To understand the expression further, we can expand it using the **distributive property** of Boolean algebra:

** (a + b)(c + d) = ac + ad + bc + bd **

This expanded form illustrates that the expression **(a + b)(c + d)** is equivalent to the **sum of all possible products** between the variables 'a', 'b', 'c', and 'd'.

### Truth Table for (a+b)(c+d)

A truth table provides a comprehensive representation of the expression's logic:

a | b | c | d | (a + b) | (c + d) | (a + b)(c + d) |
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 |

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1 | 1 | 1 | 1 | 1 | 1 | 1 |

### Applications in Digital Logic

The expression **(a + b)(c + d)** finds application in various digital logic circuits, such as:

**Multiplexers**: Used to select one of several input signals based on a control signal.**Demultiplexers**: Used to distribute a single input signal to one of several outputs.**Logic Gates**: Used to implement basic logic functions like AND, OR, NOT, XOR, etc.

By understanding the logic behind **(a + b)(c + d)** and its expansion, one can effectively analyze, design, and implement digital circuits. It forms the basis of many more complex Boolean expressions, leading to the creation of sophisticated digital systems.