Minterms and maxterms

We can obtain the canonical expression for a given function applying minterms or maxterms.

$$\begin{array}{cccccc}{x}_1& x_2& x_3& f(x_1,x_2,x_3)& \text{Minterm}& \text{Maxterm}\\ 0& 0& 0& 0& m_0={\bar{x}}_1{\bar{x}}_2{\bar{x}}_3& M_0=x_1+x_2+x_3\\ 0& 0& 1& 1& m_1={\bar{x}}_1{\bar{x}}_2{x}_3& M_1=x_1+x_2+{\bar{x}}_3\\ 0& 1& 0& 0& m_2={\bar{x}}_1{x}_2{\bar{x}}_3& M_2=x_1+{\bar{x}}_2+x_3\\ 0& 1& 1& 0& m_3={\bar{x}}_1{x}_2{x}_3& M_3=x_1+{\bar{x}}_2+{\bar{x}}_3\\ 1& 0& 0& 1& m_4=x_1{\bar{x}}_2{\bar{x}}_3& M_4={\bar{x}}_1+x_2+x_3\\ 1& 0& 1& 1& m_5=x_1{\bar{x}}_2{x}_3& M_5={\bar{x}}_1+x_2+{\bar{x}}_3\\ 1& 1& 0& 1& m_6=x_1{x}_2{\bar{x}}_3& M_6={\bar{x}}_1+{\bar{x}}_2+x_3\\ 1& 1& 1& 0& m_7=x_1{x}_2{x}_3& M_7={\bar{x}}_1+{\bar{x}}_1+{\bar{x}}_3\end{array}$$

Given the function $f(x_1,x_2,x_3)$ above, we can obtain its canonical expression in two ways:

$$\begin{array}{c}f(x_1,x_2,x_3)=\sum (m_1,m_4,m_5,m_6)\\ f(x_1,x_2,x_3)=\Pi (m_0,m_2,m_3,m_7)\end{array}$$

Example:

\hline x_1 & x_2 & f(x_1,x_2) \\ \hline 0&0&1\\ \hline 0&1&1\\ \hline 1&0&0\\ \hline 1&1&1\\ \hline \end{array}\\ $$\begin{array}{c}f(x_1,x_2)={\bar{x}}_1{\bar{x}}_2+{\bar{x}}_1{x}_2+x_1{x}_2\\ f(x_1,x_2)={\bar{x}}_1({\bar{x}}_2{x}_2)+x_1{x}_2\\ f(x_1,x_2)={\bar{x}}_1+x_1{x}_2\\ f(x_1,x_2)={\bar{x}}_1+x_2\end{array}$$