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I am trying to solve the following combination,

A | B | Cin | Sum | Cout
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1

Now the equation of Sum using SOP becomes,

ab'Cin'+ a'b'Cin + abCin + a'bCin'

Its been solved to,

a XOR b XOR Cin 

in the solution. Please guide me how can I end up with the result above? Thanks in advance.

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1 Answer

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$$ab'C_{in}'+ a'b'C_{in} + abC_{in} + a'bC_{in}'$$ $$C_{in}'(ab'+a'b) + C_{in}(a'b'+ab)$$ This is of the form: $$X'Y + XY'$$ Which is: $$X \oplus Y$$ where, $$X = C_{in}$$ $$Y = (ab'+a'b) = (a \oplus b)$$

So finally you arrive at: $$ a \oplus b \oplus C_{in}$$

Hope the answer is clear !!

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