How get I get to here C i+1 = X i Y i + C i (X i + Y i ) Not xor inside based on

Question

How get I get to here Ci+1 = XiYi + Ci(Xi + Yi) Not xor inside based on this turth table X Y C Ci+1 Si 0 0 0   0    0 0 0  1   0     1 0 1   0  0    1 0 1   1  1    0 1 0   0   0   1 1   0   1   1 0 1   1   0   1 0 1   1   1   1  1 thank you How get I get to here Ci+1 = XiYi + Ci(Xi + Yi) Not xor inside based on this turth table X Y C Ci+1 Si 0 0 0   0    0 0 0  1   0     1 0 1   0  0    1 0 1   1  1    0 1 0   0   0   1 1   0   1   1 0 1   1   0   1 0 1   1   1   1  1 thank you 0 0 0   0    0 0 0  1   0     1 0 1   0  0    1 0 1   1  1    0 1 0   0   0   1 1   0   1   1 0 1   1   0   1 0 1   1   1   1  1 thank you

Explanation / Answer

From the truth table the expression for the carry output isgiven by Ci+1 = m3 + m5 +m6 + m7
Ci+1 = X'YCi + XY'Ci +XYCi' + XYCi Ci+1 = X'YCi + XY'Ci + XY[Ci' + Ci]      Using the rule A+A'=1 Ci+1 = X'YCi + XY'Ci +XY Ci+1 = X'YCi + X [ Y'Ci + Y]         Using therule A+A'B=A+B Ci+1 = X'YCi + X [ Ci + Y] Ci+1 = X'YCi + XCi + XY Ci+1 = Ci [ X'Y+X] + XY               Usingthe rule A+A'B=A+B Ci+1 = Ci[ Y+X] + XY

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