Question

Find the Boolean product of and, where $\mathit{A}\mathbf{=}\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 1\\ 1& 1& 1& 1\end{array}\right]$ and$\mathit{B}\mathbf{=}\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\\ 1& 0\end{array}\right]$

Short answer

The Boolean product A and B of is$\left[\begin{array}{cc}1& 0\\ 1& 1\\ 1& 1\end{array}\right]$

Step-by-step solution

Step 1

$\begin{array}{r}A◻B=\left[\begin{array}{llll}1& 0& 0& 1\\ 0& 1& 0& 1\\ 1& 1& 1& 1\end{array}\right]◻\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\\ 1& 0\end{array}\right]\\ =\left[\begin{array}{cc}(1\wedge 1)\vee (0\wedge 0)\vee (0\wedge 1)\vee (1\wedge 1)& (1\wedge 0)\vee (0\wedge 1)\vee (0\wedge 1)\vee (1\wedge 0)\\ (0\wedge 1)\vee (1\wedge 0)\vee (0\wedge 1)\vee (1\wedge 1)& (0\wedge 0)\vee (1\wedge 1)\vee (0\wedge 1)\vee (1\wedge 0)\\ (1\wedge 1)\vee (1\wedge 0)\vee (1\wedge 1)\vee (1\wedge 1)& (1\wedge 0)\vee (1\wedge 1)\vee (1\wedge 1)\vee (1\wedge 0)\end{array}\right]\\ =\left[\begin{array}{c}1\vee 0\vee 0\vee 10\vee 0\vee 0\vee 0]\\ 0\vee 0\vee 0\vee 10\vee 1\vee 0\vee 0\\ 1\vee 0\vee 1\vee 10\vee 1\vee 1\vee 0\end{array}\right]\\ =\left[\begin{array}{ll}1& 0\\ 1& 1\\ 1& 1\end{array}\right]\\ \end{array}$