[Solved]: Minimizing the full adder - where did this XOR come from?

Problem Detail: 

When minimizing the full adder, I don't understand why $A(\bar{B}\bar{C} + BC)$ reduces to $A\overline{(B\oplus{C})}.$

$(\bar{B}\bar{C} + BC)\to (B\oplus{C})$ is partially decipherable, but why is $(B\oplus{C})$ inverted to $\overline{(B\oplus{C})}?$

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Full adder simplification:

$ \bar{A}\bar{B}C + \bar{A}B\bar{C} + A\bar{B}\bar{C} + ABC \\ = \bar{A}(\bar{B}C + B\bar{C}) + A(\bar{B}\bar{C} + BC) \\ = \bar{A}(B\oplus{C}) + A(\overline{B\oplus{C}}) \\ = A\oplus{(B\oplus{C})} $

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Could you help me out?

PS: I hope that this is the correct subforum of StackExchange to ask this (perhaps Electrical Engineering is the proper venue). I couldn't find appropriate tags on either site.

Asked By : Tyler

Answered By : Paresh

Hint 1: Intuitively, $\oplus$ implies exactly one of the two inputs is $1$ ($B$ and $C$ here). Whereas, $(\bar{B}\bar{C} + BC)$ implies both inputs are $0$ or both are $1$.

Hint 2: Start from $\overline{(B \oplus C)}$, expand it, use De'Morgan's laws, simplify and you should reach $(\bar{B}\bar{C} + BC)$.

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/9866