Answer to Question #101669 in Discrete Mathematics for eyks

Question

2. Produce truth tables for given Boolean expressions.

i. A̅ B̅ C+A B̅ C̅ + ABC +A̅ B C̅
ii. (A+B̅+C)(A+B+C)(A̅+B+C̅)

Answer 1

i)

$W= \overline{A}\;\overline{B}\;C, \; X= A\; \overline{B} \; \overline{C}, \; Y= A\;B\;C , \; Z= \overline{A}\; B\; \overline{C}$

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c:c} A & B & C & W& X& Y & Z & W+X+Y+Z\\ \hline 0 & 0 & 0 & 0 &0&0&0&0\\ \hdashline 0 & 0 & 1 & 1 &0&0&0&1\\ \hdashline 0 & 1 & 0 & 0 &0&0&1&1\\ \hdashline 0 & 1 & 1 & 0 &0&0&0&0\\ \hdashline 1 & 0 & 0 &0 &1&0&0&1\\ \hdashline 1 & 0 & 1&0 &0&0&0&0\\ \hdashline 1 & 1 & 0 & 0 &0&0&0&0\\ \hdashline 1 & 1 & 1 & 0 &0&1&0&1\\ \end{array}$

ii)

$X=A+\overline{B} +C, \; Y=A+B+C,\; Z=\overline{A}+B+\overline{C}$

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c} A & B & C & X& Y & Z &XYZ\\ \hline 0 & 0 & 0 & 1 &0&1&0 \\ \hdashline 0 & 0 & 1 & 1 &1&1&1\\ \hdashline 0 & 1 & 0 & 0 &1&1&0 \\ \hdashline 0 & 1 & 1 & 1 &1&1&1\\ \hdashline 1 & 0 & 0 &1 &1&1&1\\ \hdashline 1 & 0 & 1&1 &1&0&0\\ \hdashline 1 & 1 & 0 & 1 &1&1&1\\ \hdashline 1 & 1 & 1 & 1 &1&1&1\\ \end{array}$