Final Answer:
The relations r₁ and r₂ on set A are represented by the matrices:
r₁ = [0 1; 1 0] and r₂ = [1 0; 0 1].
Step-by-step explanation:
In mathematics, relations on a set can be depicted using matrices, particularly when dealing with binary relations. The matrices r₁ and r₂ signify relations on set A. Each element in the matrices corresponds to an ordered pair within the relations. For example, in matrix r₁, the element in the first row and first column is 0, signifying that the ordered pair (1,1) is not part of the relation, while the element in the first row and second column is 1, indicating that the ordered pair (1,2) is in the relation.
Likewise, in matrix r₂, the diagonal elements are both 1, representing that both (1,1) and (2,2) are in the relation, while the off-diagonal elements are 0, implying that (1,2) and (2,1) are not included in the relation.
Matrices offer a concise representation of relations, particularly in graph theory and discrete mathematics, where relations between elements or vertices are fundamental. Understanding how matrices represent relations aids in analyzing the properties and characteristics of relations, such as reflexivity, symmetry, and transitivity, by examining the patterns within these matrices.