Let $A=\begin{pmatrix}1&0&1&0\0&1&0&1\1&0&0&1\end{pmatrix}$, $B=\begin{pmatrix}0&1&0&1\1&0&1&0\1&0&0&1\end{pmatrix}$, $C=\begin{pmatrix}1&1&0&1\0&1&1&0\1&1&1&1\end{pmatrix}$ be any three boolean matrices of the same type. Find (i) $A\vee B$ (ii) $A\wedge B$ (iii) $(A\vee B)

Question

Let $A=\begin{pmatrix}1&0&1&0\0&1&0&1\1&0&0&1\end{pmatrix}$, $B=\begin{pmatrix}0&1&0&1\1&0&1&0\1&0&0&1\end{pmatrix}$, $C=\begin{pmatrix}1&1&0&1\0&1&1&0\1&1&1&1\end{pmatrix}$ be any three boolean matrices of the same type. Find (i) $A\vee B$ (ii) $A\wedge B$ (iii) $(A\vee B)\wedge C$ (iv) $(A\wedge B)\vee C$.

Accepted Answer

> [!TLDR]
>
> Join takes the entrywise maximum (a $1$ if either matrix has a $1$ there) and meet takes the entrywise minimum (a $1$ only if both matrices have a $1$ there); we compute $A\vee B$ and $A\wedge B$ first, then use them to build the last two answers.

**Step 1. Compute $A\vee B$ (entrywise max) with $A=\begin{pmatrix}1&0&1&0\0&1&0&1\1&0&0&1\end{pmatrix}$, $B=\begin{pmatrix}0&1&0&1\1&0&1&0\1&0&0&1\end{pmatrix}$.**
Row 1: $\max(1,0),\max(0,1),\max(1,0),\max(0,1)=1,1,1,1$.
Row 2: $\max(0,1),\max(1,0),\max(0,1),\max(1,0)=1,1,1,1$.
Row 3: $\max(1,1),\max(0,0),\max(0,0),\max(1,1)=1,0,0,1$.
$$A\vee B=\begin{pmatrix}1&1&1&1\1&1&1&1\1&0&0&1\end{pmatrix}.$$

**Step 2. Compute $A\wedge B$ (entrywise min).**
Row 1: $\min(1,0),\min(0,1),\min(1,0),\min(0,1)=0,0,0,0$.
Row 2: $\min(0,1),\min(1,0),\min(0,1),\min(1,0)=0,0,0,0$.
Row 3: $\min(1,1),\min(0,0),\min(0,0),\min(1,1)=1,0,0,1$.
$$A\wedge B=\begin{pmatrix}0&0&0&0\0&0&0&0\1&0&0&1\end{pmatrix}.$$

**Step 3. Compute $(A\vee B)\wedge C$ using $C=\begin{pmatrix}1&1&0&1\0&1&1&0\1&1&1&1\end{pmatrix}$ (entrywise min of $A\vee B$ and $C$).**
Row 1: $\min(1,1),\min(1,1),\min(1,0),\min(1,1)=1,1,0,1$.
Row 2: $\min(1,0),\min(1,1),\min(1,1),\min(1,0)=0,1,1,0$.
Row 3: $\min(1,1),\min(0,1),\min(0,1),\min(1,1)=1,0,0,1$.
$$(A\vee B)\wedge C=\begin{pmatrix}1&1&0&1\0&1&1&0\1&0&0&1\end{pmatrix}.$$ …

Q.Let $A=\begin{pmatrix}1&0&1&0\\0&1&0&1\\1&0&0&1\end{pmatrix}$ , $B=\begin{pmatrix}0&1&0&1\\1&0&1&0\\1&0&0&1\end{pmatrix}$ , $C=\begin{pmatrix}1&1&0&1\\0&1&1&0\\1&1&1&1\end{pmatrix}$ be any three boolean matrices of the same type. Find

Concept understanding — Binary Operations on Boolean Matrices (Join and Meet)

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