Answer to Question #263279 in Discrete Mathematics for moni

Question #263279

Obtain the sum ā€“ of ā€“ products and product - of - sum canonical form for š‘„1ā؁(š‘„2 ā‹† š‘„3 ā€² )


1
Expert's answer
2021-11-12T08:20:39-0500

Let us obtain the sum-of-products and product-of-sum canonical form for š‘„1āŠ•(š‘„2ā‹…š‘„3ā€²).š‘„_1\oplus(š‘„_2 \cdot š‘„_3'). For this let us construct the trush table:


x1x2x3x3ā€²x2ā‹…x3ā€²š‘„1āŠ•(š‘„2ā‹…š‘„3ā€²)000100001000010111011000100101101001110110111001\begin{array}{||c|c|c||c|c|c||} \hline\hline x_1 & x_2 & x_3 & x_3' & x_2\cdot x_3' &š‘„_1\oplus(š‘„_2 \cdot š‘„_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:


x1ā€²x2x3ā€²+x1x2ā€²x3ā€²+x1x2ā€²x3+x1x2x3;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:


(x1+x2+x3)(x1+x2+x3ā€²)(x1+x2ā€²+x3ā€²)(x1ā€²+x2ā€²+x3).(x_1+x_2+x_3)(x_1+x_2+x_3')(x_1+x_2'+x_3')(x_1'+x_2'+x_3).



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