page_title: Subtractor
section_title: Half subtractor
hierarchical_section_title: Subtractor / Half subtractor
caption_attribution_description: English: Logic diagram for a half subtractor.
context_page_description: In electronics, a subtractor can be designed using the same approach as that of an adder. The binary subtraction process is summarized below. As with an adder, in the general case of calculations on multi-bit numbers, three bits are involved in performing the subtraction for each bit of the difference: the minuend, subtrahend, and a borrow in from the previous bit order position. The outputs are the difference bit and borrow bit. The subtractor is best understood by considering that the subtrahend and both borrow bits have negative weights, whereas the X and D bits are positive. The operation performed by the subtractor is to rewrite as the sum.
  Subtractors are usually implemented within a binary adder for only a small cost when using the standard two's complement notation, by providing an addition/subtraction selector to the carry-in and to invert the second operand.
context_section_description: The half subtractor is a combinational circuit which is used to perform subtraction of two bits. It has two inputs, the minuend  and subtrahend  and two outputs the difference  and borrow out . The borrow out signal is set when the subtractor needs to borrow from the next digit in a multi-digit subtraction. That is,  when . Since  and  are bits,  if and only if  and . An important point worth mentioning is that the half subtractor diagram aside implements  and not  since  on the diagram is given by
  .
  This is an important distinction to make since subtraction itself is not commutative, but the difference bit  is calculated using an XOR gate which is commutative.
  The truth table for the half subtractor is:
  Using the table above and a Karnaugh map, we find the following logic equations for  and :
  .
  Consequently, a simplified half-subtract circuit, advantageously avoiding crossed traces in particular as well as a negate gate is:
        X ── XOR ─┬─────── |X-Y|,  is 0 if X equals Y, 1 otherwise
           ┌──┘   └──┐  
        Y ─┴─────── AND ── borrow, is 1 if Y > X, 0 otherwise
   
  where lines to the right are outputs and others (from the top, bottom or left) are inputs.
caption: Logic diagram for a half subtractor
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