Question

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

a) $\mathbf{A\nu B}$.

b) $\mathbf{A\Lambda B}$.

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

Short answer

a) The value of$\mathit{A}\mathbf{\vee }\mathit{B}$ is$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

(b) The value of $\mathit{A}\mathbf{\wedge }\mathit{B}$is $\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$

(c) The value of $\mathit{A}\mathbf{\odot }\mathit{B}$is $\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$

Step-by-step solution

Step 1

Given:

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

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

$\mathit{A}\mathbf{}\mathit{\nu }\mathbf{}\mathit{B}=\left[\begin{array}{cc}1\nu 0& 1\nu 1\\ 0\nu 1& 1\nu 0\end{array}\right]$

$b_1\vee b_2$is equal to 1 if$b_1=1$ or$b_2=1$ (else$b_1\vee b_2$ is equal to 0).

$=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

Step 2:

(b)

$\mathit{A}\mathbf{\wedge }\mathit{B}$takes the conjunction of each element of A with the corresponding element of B.

$\mathit{A}\mathbf{}\mathit{\Lambda }\mathbf{}\mathit{B}=\left[\begin{array}{cc}1\Lambda 0& 1\Lambda 1\\ 0\Lambda 1& 1\Lambda 0\end{array}\right]$

$b_1=b_2$is equal to 1 if$b_1=b_2$ (else$b_1\Lambda b_2$ is equal to 0 ).

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

Step 3:

(c) The ijth element of the Boolean product$A\odot B$ takes the disjunction of all conjunctions of corresponding elements A and B in the th row of and in the th column of B.

$\mathbf{A}\odot \mathbf{B}=\left[\begin{array}{l}(1\wedge 0)\vee (1\wedge 1)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(1\wedge 1)\vee (1\wedge 0)\\ (0\wedge 0)\vee (1\wedge 1)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(0\wedge 1)\vee (1\wedge 0)\end{array}\right]$

$b_1=b_2$is equal to 1 if$b_1=b_2=1$else $b_1\Lambda b_2$is equal to 0.

$=\left[\begin{array}{cc}0v1& 1v0\\ 0v1& 0v0\end{array}\right]$

is equal to 1 if or

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