# Full Adder in Digital Logic

Full Adder is the adder which adds three inputs and produces two outputs. The first two inputs are A and B and the third input is an input carry as C-IN. The output carry is designated as C-OUT and the normal output is designated as S which is SUM.

A full adder logic is designed in such a manner that can take eight inputs together to create a byte-wide adder and cascade the carry bit from one adder to the another.

**Full Adder Truth Table:**

**Logical Expression for SUM:**

= A’ B’ C-IN + A’ B C-IN’ + A B’ C-IN’ + A B C-IN

= C-IN (A’ B’ + A B) + C-IN’ (A’ B + A B’)

= C-IN XOR (A XOR B)

= (1,2,4,7)

**Logical Expression for C-OUT:**

= A’ B C-IN + A B’ C-IN + A B C-IN’ + A B C-IN

= A B + B C-IN + A C-IN

= (3,5,6,7)

**Another form in which C-OUT can be implemented:**

= A B + A C-IN + B C-IN (A + A’)

= A B C-IN + A B + A C-IN + A’ B C-IN

= A B (1 +C-IN) + A C-IN + A’ B C-IN

= A B + A C-IN + A’ B C-IN

= A B + A C-IN (B + B’) + A’ B C-IN

= A B C-IN + A B + A B’ C-IN + A’ B C-IN

= A B (C-IN + 1) + A B’ C-IN + A’ B C-IN

= A B + A B’ C-IN + A’ B C-IN

= AB + C-IN (A’ B + A B’)

Therefore COUT = AB + C-IN (A EX – OR B)

Full Adder logic circuit.

**Implementation of Full Adder using Half Adders**

2 Half Adders and a OR gate is required to implement a Full Adder.

With this logic circuit, two bits can be added together, taking a carry from the next lower order of magnitude, and sending a carry to the next higher order of magnitude.

**Implementation of Full Adder using NAND gates:**

**Implementation of Full Adder using NOR gates:**

Total 9 NOR gates are required to implement a Full Adder.

This article is contributed by **Sumouli Choudhury**

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