The complement of a function expressed as the sum of minterms equals the sum of minterms

**missing from the original function.**This is because the original function is expressed by those minterms which make the function equal to 1, whereas its complement is a expressed by those minterms for which the function is equal to 0.

So, F(A, B, C) = ∑(1,4,5,6,7)

This function has a complement

**F'(A, B, C) = ∑(0,2,3) = m**_{0}+ m_{2}+ m_{3}**Relationship between minterm and maxterm:**If we take the complement of F' by Demorgans theorem;

F = (m

_{0}+ m_{2}+ m_{3})'F = m'

_{0}. m'_{2}. m'_{3}F = M

_{0}. M_{2}. M_{3}**F = ∏(0, 2, 3)****So we conclude that, m'**_{j}= M_{j}

**General Conversion procedure - Conversion from one canonical form to another.**

Interchange the symbols ∑ and ∏

List those numbers missing from original missing from original form

To find the missing terms we consider that the total number of minterms or maxterms is 2

^{n}where 'n' is the number of binary variables in the function.

**Solve the conversion from minterm to maxterm: F = xy + x'z**

We obtain the truth table as

We use the truth table, to express 'F' as sum of minterms:

F(x,y,z) = ∑ (1,3,6,7)

F(x,y,z) = ∏ (0,2,4,5) (As total minterms are 2

^{3}= 8)

x |
y |
z |
F |

0 | 0 | 0 | 0 |

0 | 0 | 1 | 1 |

0 | 1 | 0 | 0 |

0 | 1 | 1 | 1 |

1 | 0 | 0 | 0 |

1 | 0 | 1 | 0 |

1 | 1 | 0 | 1 |

1 | 1 | 1 | 1 |

The

**Sum Of Products**is a Type of Standard form. It is a Boolean expression containing**AND**terms called product terms with one or more literals each. The sum denotes the**ORing**of these terms.**Example:**F_{1}= y' + xy + x'yz'A

**Product of Sums**is a Boolean expression containing**OR**terms called**sum terms**. Each term may have any number of literals. The product denotes the ANDing of these terms.**Example:**F_{1}= x.(y'+z).(x'+y+z')

There are 16 possible functions which are used in boolean algebra which include AND, OR.

Two functions

**Inhibition and implication**are not commutative or associative and thus are impractical to use as standard logic gates. Eight are used as standard gates in digital design, these are**Complement, Transfer, AND, OR, NAND, NOR, Exclusive OR and Equivalence**.A

**buffer**produces the transfer function but does not produce a logic operation since the binary value of the output is equal to the binary value of the input. This circuit is used for**power amplification**of the signal and is equivalent to two inverters connected in cascade.All gates shown above can be extended for more than two inputs except

**Buffer and inverter**A gate can be extended to have multiple inputs if the binary operation it represents is commutative and associative.

Boolean functions |
Operator symbols |
Name |
Comments |

F_{0} = 0 | - | Null | Binary constant 0 |

F_{1} = xy | x.y | AND | x and y |

F_{2} = xy' | x/y | Inhibition | x but not y |

F_{3} = x | - | Transfer | x |

F_{4} = x'y | y/x | Inhibition | y but not x |

F_{5} = y | - | Transfer | y |

F_{6} = xy' + x'y | x⊕y | Exclusive or | x or y but not both |

F_{7} = x + y | x+y | OR | x or y |

F_{8} = (x+y)' | - | NOR | NOT OR |

F_{9} = xy + x'y' | (x⊕y)' | equivalence | x equals y |

F_{10} = y' | y' | complement | not y |

F_{11} = x + y' | x⊂y | implication | if x then y |

F_{12} = x' | x' | complement | not x |

F_{13} = x' + y | x⊃y | implication | if x then y |

F_{14} = (xy)' | - | NAND | Not-AND |

F_{15} = 1 | - | Identity | Binary constant 1 |

The NAND and NOR functions are commutative and so their gates can be extended to have more than two inputs, however they are not associative.

**Exclusive-OR**is an odd function which means it is equal to 1 if the input variables have an odd number of 1's.**An integrated circuit**is a silicon semiconductor crystal (chip) containing the electronic components for constructing digital gates.Digital IC's are categorized by number of logic gates in the single package:

**Small scale integration:**Number of gates is fewer than 10.**Medium scale integration:**Number of gates is 10 - 1000.**Large scale integration:**Number of gates is in 1000's.**Very large scale integration:**Number of gates is in 100000's.

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