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An elementary electronic circuit in the arithmetic logic unit (ALU) of the CPU that adds one bit to another. The output of the half-adder is a result of 0 or 1 and a carry of 0 or 1.
The adder looks complicated, but it really is just a half-adder and full-adder piped together in exactly the same way it would be wired up with CMOS or TTL gates. The video below shows it in action.