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- 1. HALF ADDERAdding two single-bit binary values, X, Y produces a sum S bit and a carry out C-out bit.This operation is called half addition and the circuit to realize it is called a half adder.Half Adder Truth Table S(X,Y) = Σ (1,2) Inputs Outputs S = X’Y + XY’ X Y S C-out S = X⊕Y 0 0 0 0 0 1 1 0 C-out(x, y, C-in) = Σ (3) 1 0 1 0 C-out = XY 1 1 0 1 X Sum S Y X Half S Y Adder C-OUT C-out
- 2. FULL ADDER•Adding two single-bit binary values, X, Y with a carry input bit C- in produces a sum bit S and a carry out C-out bit. Sum S X XY C-in 00 01 11 10 0 2 6 4 Full Adder Truth Table 0 1 1 1 3 7 5 Inputs Outputs 1 1 1 C-in X Y C-in S C-out Y 0 0 0 0 0 S = X’Y’(C-in) + XY’(C-in)’ + XY’(C-in)’ + XY(C-in) S = X ⊕ Y ⊕ (C-in) 0 0 1 1 0 0 1 0 1 0 Carry C-out X 0 1 1 0 1 XY 1 0 0 1 0 C-in 00 01 11 10 0 2 6 4 1 0 1 0 1 0 1 1 1 0 0 1 1 3 7 5 1 1 1 1 C-in 1 1 1 1 1 Y S(X,Y, C-in) = Σ (1,2,4,7) C-out = XY + X(C-in) + Y(C-in) C-out(x, y, C-in) = Σ (3,5,6,7)
- 3. FULL ADDER CIRCUIT USING AND-OR X’ X’Y’C-in X Y’ X X’ C-in X’ Y X’YC-in’ Sum S Y C-in’ Y Y’ X Y C-in C-in’ XY’C-in’ C-in C-in’ X Y C-in’ XYC-in X Y X XY Y FullC-out C-in X XC-in Adder C-out C-in Y S C-in YC-in
- 4. FULL ADDER CIRCUIT USING XOR X Y Sum S X Y C-in Full X XYC-out C-in Adder Y X XC-in C-out S C-in Y C-in YC-in
- 5. HALF SUBTRACTORSubtracting a single-bit binary value Y from anther X (I.e. X -Y ) produces a difference bit D and a borrow out bit B-out. This operation is called half subtraction and the circuit to realize it is called a half subtractor. Half Subtractor Truth Table D(X,Y) = Σ (1,2) Inputs Outputs D = X’Y + XY’ X Y D B-out D = X⊕Y 0 0 0 0 0 1 1 1 B-out(x, y, C-in) = Σ (1) 1 0 1 0 B-out = X’Y 1 1 0 0 X Difference D Y X Half D Y Subtractor B-OUT B-out
- 6. FULL SUBTRACTOR•Subtracting two single-bit binary values, Y, B-in from a single-bit value X produces a difference bit D and a borrow out B-out bit. This is called full subtraction. Difference D X XY B-in 00 01 11 10 0 2 6 4 0 1 1 Full Subtractor Truth Table 1 3 7 5 Inputs Outputs 1 1 1 B-in X Y B-in D B-out Y 0 0 0 0 0 S = X’Y’(B-in) + XY’(B-in)’ + XY’(B-in)’ + XY(B-in) S = X ⊕ Y ⊕ (C-in) 0 0 1 1 1 0 1 0 1 1 Borrow B-out X 0 1 1 0 1 XY 1 0 0 1 0 B-in 00 01 11 10 0 2 6 4 1 0 1 0 0 0 1 1 1 0 0 0 1 3 7 5 1 1 1 1 B-in 1 1 1 1 1 Y S(X,Y, C-in) = Σ (1,2,4,7) B-out = X’Y + X’(B-in) + Y(B-in) C-out(x, y, C-in) = Σ (1,2,3,7)
- 7. FULL SUBTRACTOR CIRCUIT USING AND-OR X’ X’Y’B-in X Y’ X X’ B-in X’ Y X’YB-in’ Difference D Y B-in’ Y Y’ X Y B-in B-in’ XY’B-in’ B-in B-in’ X Y B-in’ XYB-in X Y X’ X’Y Y FullB-out B-in X’ X’B-in Subtractor B-out B-in Y D B-in YB-in
- 8. FULL SUBTRACTOR CIRCUIT USING XOR X Y Difference D X Y B-in Full X’ X’YB-out B-in Subtractor Y X’ X’B-in B-out D B-in Y B-in YB-in

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