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Say we want a circuit that will add two bits, we need four rules: $$0+0=00\\ 0+1=01\\ 1+0=00\\ 1+1=10$$ Where our left-most bit in the answer will be our "carry bit" and the right-most is the "sum bit", this mirrors how you learned to do addition.
$$A+B$$CarrySum
$$0+0$$00
$$0+1$$01
$$1+0$$01
$$1+1$$11
So we build a circuit to generate this logic function: We call this a half adder because although it will generate a carry bit it can't accept one (say from a previous addition from less significant bits in the same multi-bit addition).

In order to add two multi-bit binary numbers we need an adder circuit that accepts a carry bit that might have been generated from an earlier addition. Now we don't wire this up every time we want something added, you can buy an adder chip that implements this for you.
A full-adder chip accepts two input bits and a bit which represents an input carry (whether a carry was generated in the previous btis' addition) and outputs a carry bit and a sum bit. Its symbol is: Now we can use these chips to compute multi-bit additions. A full adder does just what we do when we perform an addition. By linking the carry out of one chip to the carry in of the next chip carries are propogated through the addition, just like we do when adding. So to add two numbers $$A$$ and $$B$$ we apply $$A_0$$ and $$B_0$$ to the first chip then attach its carry out to the next chip's carry in and apply $$A_1$$ and $$B_1$$ to that chip and so on.
To add two multi-bit binary numbers $$A$$ and $$B$$ attach your full-adder chips as follows: Where $$A_0$$ and $$B_0$$ are applied to the bottom chip and $$A_3$$ and $$B_3$$ are applied to the top chip. The result is read from each of the chips' sum lines with the MSB at the top.