Calculating the Boolean Product of Matrices

The question asks us to find the boolean product of two matrices, \(A\) and \(B\). The boolean product, denoted by \(A \odot B\), is similar to standard matrix multiplication but uses boolean operations (AND and OR) instead of standard multiplication and addition.

Given matrices:

Matrix \(A\) (\(4 \times 3\)):

\(A = \left[ {\begin{array}{*{20}{c}} 1&1&0\\ 0&1&0\\ 1&1&0\\ 0&0&1 \end{array}} \right]\)

Matrix \(B\) (\(3 \times 4\)):

\(B = \left[ {\begin{array}{*{20}{c}} 1&0&0&0\\ 0&1&1&0\\ 1&0&1&1\\ \end{array}} \right]\)

The boolean product \(C = A \odot B\) will be a \(4 \times 4\) matrix. Let \(C = \left[ c_{ij} \right]\). The element \(c_{ij}\) is calculated as the boolean OR of the boolean ANDs of the corresponding elements from the \(i\)-th row of \(A\) and the \(j\)-th column of \(B\). The formula for \(c_{ij}\) is:

\(c_{ij} = (a_{i1} \land b_{1j}) \lor (a_{i2} \land b_{2j}) \lor \dots \lor (a_{ik} \land b_{kj})\)

where \(\land\) represents boolean AND (multiplication, \(1 \cdot 1 = 1\), all others \(0\)) and \(\lor\) represents boolean OR (addition, \(0+0=0\), all others \(1\)). For our matrices A (4x3) and B (3x4), the resulting matrix C (4x4) element \(c_{ij}\) is:

\(c_{ij} = (a_{i1} \land b_{1j}) \lor (a_{i2} \land b_{2j}) \lor (a_{i3} \land b_{3j})\)

Let's calculate each element of the resulting matrix \(C\):

• \(c_{11} = (a_{11} \land b_{11}) \lor (a_{12} \land b_{21}) \lor (a_{13} \land b_{31}) = (1 \land 1) \lor (1 \land 0) \lor (0 \land 1) = 1 \lor 0 \lor 0 = 1\)
• \(c_{12} = (a_{11} \land b_{12}) \lor (a_{12} \land b_{22}) \lor (a_{13} \land b_{32}) = (1 \land 0) \lor (1 \land 1) \lor (0 \land 0) = 0 \lor 1 \lor 0 = 1\)
• \(c_{13} = (a_{11} \land b_{13}) \lor (a_{12} \land b_{23}) \lor (a_{13} \land b_{33}) = (1 \land 0) \lor (1 \land 1) \lor (0 \land 1) = 0 \lor 1 \lor 0 = 1\)
• \(c_{14} = (a_{11} \land b_{14}) \lor (a_{12} \land b_{24}) \lor (a_{13} \land b_{34}) = (1 \land 0) \lor (1 \land 0) \lor (0 \land 1) = 0 \lor 0 \lor 0 = 0\)
• \(c_{21} = (a_{21} \land b_{11}) \lor (a_{22} \land b_{21}) \lor (a_{23} \land b_{31}) = (0 \land 1) \lor (1 \land 0) \lor (0 \land 1) = 0 \lor 0 \lor 0 = 0\)
• \(c_{22} = (a_{21} \land b_{12}) \lor (a_{22} \land b_{22}) \lor (a_{23} \land b_{32}) = (0 \land 0) \lor (1 \land 1) \lor (0 \land 0) = 0 \lor 1 \lor 0 = 1\)
• \(c_{23} = (a_{21} \land b_{13}) \lor (a_{22} \land b_{23}) \lor (a_{23} \land b_{33}) = (0 \land 0) \lor (1 \land 1) \lor (0 \land 1) = 0 \lor 1 \lor 0 = 1\)
• \(c_{24} = (a_{21} \land b_{14}) \lor (a_{22} \land b_{24}) \lor (a_{23} \land b_{34}) = (0 \land 0) \lor (1 \land 0) \lor (0 \land 1) = 0 \lor 0 \lor 0 = 0\)
• \(c_{31} = (a_{31} \land b_{11}) \lor (a_{32} \land b_{21}) \lor (a_{33} \land b_{31}) = (1 \land 1) \lor (1 \land 0) \lor (0 \land 1) = 1 \lor 0 \lor 0 = 1\)
• \(c_{32} = (a_{31} \land b_{12}) \lor (a_{32} \land b_{22}) \lor (a_{33} \land b_{32}) = (1 \land 0) \lor (1 \land 1) \lor (0 \land 0) = 0 \lor 1 \lor 0 = 1\)
• \(c_{33} = (a_{31} \land b_{13}) \lor (a_{32} \land b_{23}) \lor (a_{33} \land b_{33}) = (1 \land 0) \lor (1 \land 1) \lor (0 \land 1) = 0 \lor 1 \lor 0 = 1\)
• \(c_{34} = (a_{31} \land b_{14}) \lor (a_{32} \land b_{24}) \lor (a_{33} \land b_{34}) = (1 \land 0) \lor (1 \land 0) \lor (0 \land 1) = 0 \lor 0 \lor 0 = 0\)
• \(c_{41} = (a_{41} \land b_{11}) \lor (a_{42} \land b_{21}) \lor (a_{43} \land b_{31}) = (0 \land 1) \lor (0 \land 0) \lor (1 \land 1) = 0 \lor 0 \lor 1 = 1\)
• \(c_{42} = (a_{41} \land b_{12}) \lor (a_{42} \land b_{22}) \lor (a_{43} \land b_{32}) = (0 \land 0) \lor (0 \land 1) \lor (1 \land 0) = 0 \lor 0 \lor 0 = 0\)
• \(c_{43} = (a_{41} \land b_{13}) \lor (a_{42} \land b_{23}) \lor (a_{43} \land b_{33}) = (0 \land 0) \lor (0 \land 1) \lor (1 \land 1) = 0 \lor 0 \lor 1 = 1\)
• \(c_{44} = (a_{41} \land b_{14}) \lor (a_{42} \land b_{24}) \lor (a_{43} \land b_{34}) = (0 \land 0) \lor (0 \land 0) \lor (1 \land 1) = 0 \lor 0 \lor 1 = 1\)

Putting these elements together, the resulting boolean product matrix \(A \odot B\) is:

\(A \odot B = \left[ {\begin{array}{*{20}{c}} 1&1&1&0\\ 0&1&1&0\\ 1&1&1&0\\ 1&0&1&1 \end{array}} \right]\)

This result matches the first option provided.

Revision Table: Boolean Matrix Operations

Additional Information on Boolean Matrices

Boolean matrices are matrices whose entries are either 0 or 1. They are often used in various applications, including representing relationships in discrete structures like directed graphs and relations.

• Representing Relations: A boolean matrix can represent a relation \(R\) between two sets \(A = \{a_1, \dots, a_m\}\) and \(B = \{b_1, \dots, b_n\}\). The matrix \(M_R\) is an \(m \times n\) matrix where \(M_R[i,j] = 1\) if \((a_i, b_j) \in R\) and \(0\) otherwise.
• Composition of Relations: The boolean product of matrices is particularly useful for finding the matrix representing the composition of two relations. If matrix \(M_R\) represents relation \(R\) from \(A\) to \(B\), and \(M_S\) represents relation \(S\) from \(B\) to \(C\), then the matrix representing the composite relation \(R \circ S\) from \(A\) to \(C\) is given by the boolean product \(M_R \odot M_S\).
• Pathfinding: Boolean matrix multiplication can also be used to determine the existence of paths of a certain length in a directed graph.