Question

Let $\mathit{A}\mathbf{=}\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]\mathbf{\text{and}}\mathit{B}\mathbf{=}\left[\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]$

Find

a) $\mathit{A}\mathbf{\vee }\mathit{B}$.

b) $\mathit{A}\mathbf{\wedge }\mathit{B}$.

c)$\mathit{A}\mathbf{\odot }\mathit{B}$.

Short answer

a) Required answer is $\mathbf{A}\vee \mathbf{B}=\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]$.

b) Required answer is $\mathbf{A}\Lambda \mathbf{B}=\left[\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right]$.

c) Required answer is$A\odot B=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 0& 1\end{array}\right]$

Step-by-step solution

Step 1:

Given: $\mathit{A}=\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 0& 0& 1\end{array}\right]\mathit{B}=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 0& 1\end{array}\right]$

(a) $\mathbf{A}\vee \mathbf{B}$takes the disjunction of each element of B with the corresponding element of.

$\mathbf{A}\vee \mathbf{B}=\left[\begin{array}{ccc}1\vee 0& 0\vee 1& 1\vee 1\\ 1\vee 1& 1\vee 0& 0\vee 1\\ 0\vee 1& 0\vee 0& 1\vee 1\end{array}\right]$

${\mathrm{b}}_1\vee {\mathrm{b}}_2$is equal to 1 if${\mathrm{b}}_1=1$ or${\mathrm{b}}_2=1$ (else 0 ).

role="math" localid="1668489394918" $A\nu B=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 0& 1\end{array}\right]$

Step 3:

(c) The${\left(ij\right)}^{th}$ element of the Boolean productrole="math" localid="1668489454199" $\mathit{A}\mathbf{\odot }\mathit{B}$ takes the disjunction of all conjunctions of corresponding elements A and B in the ith row of A and in the jth column of.

role="math" localid="1668489606325" $\begin{array}{r}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\\ \mathbf{A}\odot \mathbf{B}=\left[\begin{array}{lll}(1\wedge 0)\vee (0\wedge 1)\vee (1\wedge 1)& (1\wedge 1)\vee (0\wedge 0)\vee (1\wedge 0)& (1\wedge 1)\vee (0\wedge 1)\vee (1\wedge 1)\\ (1\wedge 0)\vee (1\wedge 1)\vee (0\wedge 1)& (1\wedge 1)\vee (1\wedge 0)\vee (0\wedge 0)& (1\wedge 1)\vee (1\wedge 1)\vee (0\wedge 1)\\ (0\wedge 0)\vee (0\wedge 1)\vee (1\wedge 1)& (0\wedge 1)\vee (0\wedge 0)\vee (1\wedge 0)& (0\wedge 1)\vee (0\wedge 1)\vee (1\wedge 1)\end{array}\right]\end{array}$

$b_1\wedge b_2$is equal to 1 if$b_1=b_2=1$

$=\left[\begin{array}{l}0\vee 0\vee 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\vee 0\vee 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\vee 0\vee 1\\ 0\vee 1\vee 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\vee 0\vee 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\vee 1\vee 0\\ 0\vee 0\vee 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\vee 0\vee 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\vee 0\vee 1\end{array}\right]$

$b_1\vee b_2\vee b_3$is equal to 1 if$b_1=1$ or$b_2=1$ or$b_3=1$

$\mathbf{=}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 0& 1\end{array}\right]$