To solve this task, you have to bear in mind the logic of a binary addition...
0  0  1  1 
+ 0  + 1  + 0  + 1 
= 00  = 01  = 01  = 10 
If you are expecting only a 1bit response, everything looks easy. All you need for this is a XOR XOR gate.
But if you are adding a 1 and 1, the answer is more complicated. A 1 must be transmitted to the next position. This bit is called the Carry bit. And as an output of a logic circuit, it is indicated by CO (Carry Out).

You realize the result for CO by using an AND gate, as you see from the truth table. The adder of two onedigit binary numbers is called a half adder. But what you need to add two 4digit numbers? 