Here is the question I'm trying to solve:
Use algebraic manipulation to find the minimum sum-of-products expression for the function $f=x_1x_3+x_1\overline{x}_2+\overline{x}_1x_2x_3+\overline{x}_1\overline{x}_2\overline{x}_3$
The way I attempted it was as follows:
f = ac + ab' + a'bc + a'b'c'
= ac + b'(a + a'c') + a'bc [distributive]
= c(a + a'b) + b'(a + a'c') [distributive]
= c(a + b) + b'(a + c') [using x + x'y = x + y]
= ac + bc + ab' + b'c'
where a = x1, b = x2, c = x3
The correct answer, however is:
$f=x_1x_3+x_2x_3+\overline{x}_2\overline{x}_3$
Can anyone tell me where I went wrong?
Best Answer
$\begin{align} &ac + bc + ab' + b'c'\\=~&bc +~ ac + ab' + b'c'\\ =~& bc+~(abc+ab'c)+(ab'c+ab'c')+(ab'c'+a'b'c')\\\vdots~~& \\ =~& bc+~ac+b'c'\\=~& ac+bc+b'c' \end{align} $
Complete the missing steps, and you are done.