Full subtractor using a 2×4 decoder

A full subtractor computes $A - B - B_{in}$, giving difference $D$ and borrow $B_{out}$. A single 2×4 decoder has only two inputs, so feed it $A$ and $B$; its outputs are the minterms of $A, B$, and $B_{in}$ is combined with gates.

Truth table grouped by AB

Equations

$$\begin{aligned} D &= (Y_1 + Y_2) \oplus B_{in} \\ B_{out} &= Y_1 + (Y_0 + Y_3)\,B_{in} \end{aligned}$$

(Here $Y_1 + Y_2 = A \oplus B$ and $Y_0 + Y_3 = A \odot B$.)

          +--------+
  A ----->| 2x4  Y0|---> E (OR)
  B ----->|      Y1|---> P (OR), Bout (OR)
          |      Y2|---> P (OR)
          |      Y3|---> E (OR)
          +--------+
  Y0, Y3 ---[OR]--- E  (= A xnor B)
  Y1, Y2 ---[OR]--- P  (= A xor B)
  P, Bin ---[XOR]------------------ D
  E, Bin ---[AND]--- E.Bin --+
  Y1 ------------------------+-[OR]-- Bout

Gates: 2 OR (2-input), 1 XOR, 1 AND, 1 OR.

Check: $A=0, B=1, B_{in}=1$: $Y_1 = 1$, $P = 1$, $D = 1 \oplus 1 = 0$, $B_{out} = 1$; indeed $0 - 1 - 1 = -2$, i.e. $D = 0$ with borrow 1. All 8 rows were verified.