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Subtractors are digital circuits that perform binary subtraction on two binary numbers. They operate on bits (0s and 1s) and produce a difference (result of subtraction) and a borrow (indication of borrowing from the next higher-order bit).
1) Half Subtractor
A half subtractor is a basic circuit that subtracts two single-bit binary numbers (A and B) and produces two outputs:
Difference (D): The result of subtracting B from A (A - B).
Borrow (Borrow): A flag indicating whether a borrow was needed from the next higher-order bit (1 if borrow occurred, 0 otherwise).
A | B | Difference (D) | Borrow (B) |
0 | 0 | 0 | 0 |
0 | 1 | 1 | 1 |
1 | 0 | 1 | 0 |
1 | 1 | 0 | 0 |
Implementation:
A half subtractor can be implemented using two basic logic gates:
XOR (Exclusive OR) gate: Calculates the difference (A - B) by performing an XOR operation on the two input bits.
AND gate: Determines the borrow by calculating the AND of the two input bits.
Circuit Diagram:
Explanation:
The XOR gate calculates the difference (D) by performing bitwise XOR on A and B.
If A is 1 and B is 0, or if A is 0 and B is 1, the XOR result is 1 (indicating a difference of 1).
If both A and B are the same (0 or 1), the XOR result is 0 (no difference).
The AND gate calculates the borrow (Borrow) by checking if both A and B are 1. This indicates that we need to borrow 1 from the next higher-order bit for subtraction.
2) Full Subtractor
A full subtractor is a more complex circuit that subtracts two single-bit binary numbers (A and B) while considering a borrow (BorrowIn) from the previous higher-order bit subtraction. It produces two outputs:
Difference (D): The final result of the subtraction (A - B - BorrowIn).
BorrowOut (Borrow): A flag indicating whether a borrow needs to be propagated to the next higher-order bit (1 if borrow occurs, 0 otherwise).
Truth Table:
A | B | Borrow in | Difference | Borrow |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 0 |
1 | 1 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
Implementation:
A full subtractor can be built using two half subtractors and an OR gate:
Half subtractor 1: Subtracts B from A, generating a difference (D1) and a borrow (Borrow1).
Half subtractor 2: Subtracts BorrowIn from D1, providing the final difference (D) and a borrow (BorrowOut).
OR gate: Combines Borrow1 and the BorrowIn to determine the BorrowOut signal sent to the next higher-order bit.
Circuit diagram :-
Expression for full subtractor:-
D(difference ) = A xor B xor Bin
B(out) = (!A and B) or ((!A or B) and Bin)
Explanation:
Half subtractor 1 performs the initial subtraction of B from A, giving a difference (D1) and a borrow (Borrow1).
Half subtractor 2 takes D
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