2. HALF ADDER
Adding 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

3. 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)

4. 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
Full
C-out C-in X
XC-in
Adder C-out
C-in
Y
S
C-in YC-in

5. FULL ADDER CIRCUIT USING XOR
X
Y Sum S
X Y C-in
Full X XY
C-out C-in
Adder Y
X
XC-in C-out
S C-in
Y
C-in YC-in

6. HALF SUBTRACTOR
Subtracting 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

7. 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)

8. 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
Full
B-out B-in X’
X’B-in
Subtractor B-out
B-in
Y
D
B-in YB-in

9. FULL SUBTRACTOR CIRCUIT USING XOR
X
Y Difference D
X Y B-in
Full X’ X’Y
B-out B-in
Subtractor Y
X’
X’B-in
B-out
D B-in
Y
B-in YB-in