Q.Let $A=\begin{bmatrix}0 & 1\\1 & 1\end{bmatrix}$, $B=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$ be any two Boolean matrices of the same type. Find $A\vee B$ and $A\wedge B$.

A Boolean matrix is a matrix all of whose entries are $0$ or $1$ (the "on/off" values used to model electrical switches, adjacency in graph theory, etc.). Two Boolean matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ of the same order combine via two new binary operations.

Join, $A\vee B$. Entrywise: $c_{ij}=a_{ij}\vee b_{ij}=\max(a_{ij},b_{ij})$ -- i.e. $c_{ij}=1$ if either $a_{ij}=1$ or $b_{ij}=1$, and $c_{ij}=0$ only when both entries are $0$.

Meet, $A\wedge B$. Entrywise: $c_{ij}=a_{ij}\wedge b_{ij}=\min(a_{ij},b_{ij})$ -- i.e. $c_{ij}=1$ only when both $a_{ij}=1$ and $b_{ij}=1$.

Which of the five properties do join and meet satisfy? Let $\mathbb B$ be the set of all Boolean matrices of a fixed order.

• Closure: since $a_{ij}\vee b_{ij}$ and $a_{ij}\wedge b_{ij}$ are each either $0$ or $1$, both $A\vee B$ and $A\wedge B$ stay in $\mathbb B$ -- both are binary operations on $\mathbb B$.
• Commutative: $\max$ and $\min$ of two numbers do not care about order, so both $\vee$ and $\wedge$ are commutative.
• Associative: both are associative, entry by entry. …