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The truth table of a full adder is the same as a binary addition of three bits.
Input Output
A B C in Sum C out
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Referring C as C, the Boolean expression of the sum term:
in
= A'.B'.C + A'.B.C' + A.B'.C' + A.B.C
which is equivalent to a three variables XOR gate.
Boolean expression for carry term = A'.B.C + A.B'.C + A.B.C' + A.B.C which on minimisation becomes A.B + B.C + C.A
[Minimisation will be covered in the class XII book]
The logic circuit diagram of a full adder circuit is:
A Sum = A⊕B⊕C in
B
C in
A.B
A.B + B.C + C .A
in
in
B.C in
C .A
in
The full adder circuit can also be represented as two half adders connected by OR gate as follows:
Sum of full adder = A'.B'.C + A'.B.C' + A.B'.C' + A.B.C
= A'.(B'.C+B.C') + A.(B'.C'+B.C) [Distributive law]
= A'.(B⊕C) + A.(B9C)
= A⊕B⊕C
Carry of full adder = A'.B.C + A.B'.C + A.B.C' + A.B.C
= C.(A'.B+A.B') + A.B.(C'+C) [Distributive law]
= C.(A⊕B ) + A.B [Complement law]
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