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The new circuit combines three inputs: two input bits and a carry.
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The gate-level description of a full adder follows:
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Combination of AND-gates is a decoder circuit for each possible combination of inputs.
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Decoder produces output of a single 1 (with the rest of 0s) for each unique combinaton of input signals.
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The OR gates re-combine unique output 1 into desired combination of carry and sum outputs.
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Note that AND-gate on top of the circuit is not connected to either of the OR-gates, because it should never produce any 1s in either output.
