What do we do when we have to create a scoreboard for a certain design logic? For a memory I understand that we can compare the data written to DUT at a certain address to the data read at the same address. Suppose for a basic example of full adder how do we maintain a scoreboard. What values do we compare and what method will we be using?
Typically scoreboard is used to compare the data as you have mentioned in your question. Every scoreboard has a reference model (just the functionality of DUT).
For a full adder, you will implement the variables(declared in scoreboard) addition and receive the values from the monitor through the interface and your reference model calculates the sum and carry. This is compared with the sum and carry coming out of DUT captured by the monitor.