How to answer matrices problems involving "meet" and "join"?

Question

Problem: Given the matrix below find the meet and join of A and B.

\begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}

\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ \end{bmatrix}

How to answer this kind of questions? I have learned the basics to advance math and know how to answer matrices involving problems, but I am new to this "meet" and "join" type of questions. Any help would be appreciated.

Answer 1

It is just the pointwise meet and join of each entry. So the $(0,0)$ entry (top left) is $1 = 0 \lor 1$ for the join-matrix and $0 \land 1 = 0$ for the meet-matrix. The $(2,2)$ entry of the meet is $1 \lor1= 1$ and of the join $1 \land 1 = 1$.

In formulae:

$$(A \land B)_{ij} = a_{ij} \land b_{ij}$$

and

$$(A \lor B)_{ij} = a_{ij} \lor b_{ij}$$

So for this case the join is

$$\begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix}$$

and the meet is $$\begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 0 & 1 \end{bmatrix}$$

Answer 2

Join two matrix is $1$ if both corresponding elements in two matrix is $1$. Otherwise $0$.

Meet of two boolean matrix is $1$ if either of corresponding element is $1$ and $0$ if both element is $0$.