The subtraction of two binary numbers may be accomplished by taking the complement of the subtrahend and adding it to the minuhend. By this method, the subtraction operation becomes an addition operation requiring full adders for its machine implementation. It is possible to implement subtraction with logic circuits in a direct manner. By this method, each subtrahend bit of the number is subtracted from its corresponding significant minuhend bit to form a different bit. If the minuhend bit is smaller than the subtrahend bit, a 1 is borrowed from the next significant position. The fact that a 1 has been borrowed must be conveyed to the next higher pair of bits by means of a binary signal coming out (output) of a given stage and going into (input) the next higher stage. It is same for the half-adder and full-adder, half-subtractor and full-subtractor circuits.

A full-subtractor is a combinational circuit that performs a subtraction between two bits, taking into account that a 1 may have been borrowed by a lower significant stage. The circuit has three inputs and two outputs. The three inputs is denoted by a, b and c which represent the minuhend, subtrahend and the previous borrow bit respectively.

Truth Table

Input a | Input b | Input c | Output B | Output D |

0 | 0 | 0 | 0 | 0 |

0 | 0 | 1 | 1 | 1 |

0 | 1 | 0 | 1 | 1 |

0 | 1 | 1 | 1 | 0 |

1 | 0 | 0 | 0 | 1 |

1 | 0 | 1 | 0 | 0 |

1 | 1 | 0 | 0 | 0 |

1 | 1 | 1 | 1 | 1 |

Solution Using K-Map

For D

a\bc 00 01 11 10

1 | 1 | ||

1 | 1 |

D=a'b'c+a'bc'+ab'c'+abc

For B

a\bc 00 01 11 10

1 | 1 | 1 | |

1 |

Circuit Diagram

The simplified boolean functions for the two oututs of the full subtractor are derived from the K-maps and can be represented as

D=a'b'c+a'bc'+ab'c'+abc

B=a'b+a'c+bc

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