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I'm very confused about the reasoning for these circuits being called 'full adders' and 'half adders'

I've read before that 'half adders' are called so, because two of them make up a 'full adder', and a 'full adder' is 'full' because you can get all the values $(0,0), (0,1), (1,0), (1,1)$.

But what irks me is that adding two bits would never get you $(1,1)$ in the first place, so why not call it a 2-bit adder, since that's exactly what it does. By extension, a 'full adder', in my mind, should be called a 3-bit adder.

If i think of full adders as '3 bit adders', would I ever run into any conceptual troubles? Same goes with '2 bit adders' and half-adders. Are there any historical insights as to why they are called such?

Cheers.

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A "half adder" only does half the job: It adds two inputs x and y.

A "full adder" does the full job: Most of the time, you need to process two inputs, plus a carry from another (half or full adder). A "full adder" adds three inputs x, y and z.

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