Final answer:
The student's question addresses the characteristics of a binary relation defined by y² = x² over the set of real numbers. This binary relation includes all pairs (x, y) where y can be either the positive or negative square root of x, and this relation is represented on a two-dimensional (x-y) graph.
Step-by-step explanation:
The question appears to be about understanding the nature of a particular binary relation over the set of real numbers (R). A binary relation R on sets X and Y is a collection of ordered pairs (x, y) where x belongs to X and y belongs to Y. In the context of this problem, we might define a binary relation based on the equation y² = x², which implies that y could be either the positive or negative square root of x. It's important to note that y² = x² will be satisfied for every x and y that are either both positive or both negative. Moreover, when x = y = 0, the origin is included in this relation, since 0 is both a positive and negative square root of itself.
Typically, when dealing with square roots in equations, it is assumed that they return the principal (non-negative) root. However, y² = x² does not make this assumption, as both positive and negative roots satisfy the equation. Therefore, the binary relation defined by this equation includes both (x, y) and (x, -y) for all x in the real numbers, except at the origin where there's only one pair (0, 0). This kind of relation is often represented visually via two-dimensional (x-y) graphing.
When discussing properties like these in relation to quadratic equations, it's noted that equations based on physical data typically have real roots. This ties into the understanding that our binary relation in question would always result in real numbers y for every real number x.