How does the value matrix correspond to the truth table of an operation?

Question

I have been self studying mathematical logic through A Concise Introduction to Mathematical Logic by Wolfgang Rautenberg and got stuck. I am unable to understand how the value matrix corresponds with the truth table of a given operation for instance the AND operation. I know that in a matrix the place an element is placed into important. Is there some kind of relation based on the position in the matrix. For example the ith column and jth row. Why are the values ordered as they are within the matrix?

Answer 1

Looks like the rows,1,2 go with the truth values, T,F of p, and columns 1,2 go with the truth values T,F of q in the connective p (connecvtive) q.

Thus row 1 column 2 corresponds to p true (T) and q false (F), etc.

Answer 2

The author is using an abbreviated form of the notation often used for binary operations on finite algebraic structures, leaving out the row and column labels. Drawn up in full, the table for conjunction would look like this: $$ \begin{array}{c|cc} {\land} & 1 & 0 \\\hline 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} $$

The idea is that to evaluate $a \land b$, you look for the row labelled $a$ and the column labelled $b$ and the result is the corresponding entry in the matrix: $1 \land 1 = 1$, $1 \land 0 = 0$, $0 \land 1 = 0$ and $0 \land 0 = 0$.