Answer to Question #263279 in Discrete Mathematics for moni

Question

Obtain the sum – of – products and product - of - sum canonical form for 𝑥1⨁(𝑥2 ⋆ 𝑥3 ′ )

Answer 1

Let us obtain the sum-of-products and product-of-sum canonical form for $𝑥_1\oplus(𝑥_2 \cdot 𝑥_3').$ For this let us construct the trush table:

$\begin{array}{||c|c|c||c|c|c||} \hline\hline x_1 & x_2 & x_3 & x_3' & x_2\cdot x_3' &𝑥_1\oplus(𝑥_2 \cdot 𝑥_3')\\ \hline\hline 0 & 0 & 0 & 1 & 0 & 0\\ \hline 0 & 0 & 1 & 0 & 0 & 0\\ \hline 0 & 1 & 0 & 1 & 1 & 1\\ \hline 0 & 1 & 1 & 0 & 0 & 0\\ \hline 1 & 0 & 0 & 1 & 0 & 1\\ \hline 1 & 0 & 1 & 0 & 0 & 1\\ \hline 1 & 1 & 0 & 1 & 1 & 0\\ \hline 1 & 1 & 1 & 0 & 0 &1\\ \hline\hline \end{array}$

It follows from the trush table that the sum-of-products canonical form is the following:

$x_1'x_2x_3'+x_1x_2'x_3'+x_1x_2'x_3+x_1x_2x_3;$

and the product-of-sum canonical form is the following:

$(x_1+x_2+x_3)(x_1+x_2+x_3')(x_1+x_2'+x_3')(x_1'+x_2'+x_3).$