The carry bit output is given by the relationship

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The full adder as a logical unit must obey the truth table at left.

This truth table translates to the logical relationship

which when simplified can be expressed as

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The carry bit output is given by the relationship

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The full adder can be implemented with six gates as shown at left.

The logical relationship for the sum is

which when simplified can be expressed as

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The carry bit output is given by the relationship

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The full adder logic relationship from the truth table at left is:

If

then this can be seen to be another XOR statement,

which can be written as

The sum is given by the relationship

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Adding digits in binary numbers with the full adder involves handling the "carry" from one digit to the next. From the truth table at left the logic relationship can be seen to be

The basic identity X+X=X can be used for simplification where X = ABC. Since we have an X, we can throw two more "OR X" 's without changing the logic, giving

This simplifies to

using

, etc.