In a full adder, the Carry-out output (Cout) is calculated using the formula Cout = (A.B) + (Cin.(A XOR B)). What is the logical reasoning behind this formula?

This formula breaks down into two separate ways a carry-out can be generated. First term, (A.B): if both A and B are 1, that alone guarantees a carry-out is generated, regardless of Cin (since 1+1=10 in binary already produces a carry on its own). Second term, (Cin.(A XOR B)): this covers the case where exactly one of A or B is 1 (meaning A XOR B = 1, so A and B 'differ'), and additionally there's an incoming carry (Cin=1) -- in that case, you're effectively adding 1+0+1 or 0+1+1, both of which equal 2 in decimal (binary '10'), which does produce a carry. OR-ing these two conditions together with a + gives you every situation where a carry-out should occur, covering all cases where the true sum of A+B+Cin is 2 or 3 (both of which require a carry).

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