How does it evaluate A XOR B XOR C?

Question

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.

Answer 1

$$ab'C_{in}'+ a'b'C_{in} + abC_{in} + a'bC_{in}'$$ $$C_{in}'(ab'+a'b) + C_{in}(a'b'+ab)$$ 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 !!