Definition – Relation. – Math-reference.net

Contents
1. Definition
2. Geometric meaning

Definition

Let $S$ be a set. We define a relation on $S$ to be a subset, say $S'$ , of the cross product of $S$ with iteself. Ie, a relation $R$ is defined as

$R \subseteq{S\times S}$

Alternatively, we can also define

Geometric meaning

The definition simply states that, given a linear combination of two elements of $S$ , the resulting vector is also in $S$ . This definition relates to lines and planes in $mathbb{R}^2$ and $mathbb{R}^3$ . To see why this is true, we consider two cases. The first is that $mathbf{x}$ and $mathbf{y}$ are scalar multiples of each other, say $mathbf{x} = c times_2 mathbf{y}$ . Then, we have

$begin{aligned} atimes_{2}mathbf{x} + btimes_{2}(atimes_{2}mathbf{x}) &= (1+b)times_{2}(atimes_{2}mathbf{x}) \ &=[(1+b)times_{1}(a)]times_{2}mathbf{x} end{aligned}$

Which is simply a constant times a vector, which is the equation of a straight line in $mathbb{R}^2$ and $mathbb{R}^3$ . Now, if the two vectors are not scalar multiples of each other, then we have the equation of a plane, which applies in both $mathbb{R}^2$ and $mathbb{R}^3$ . To see this, imagine we were to fix $a$ in $[atimes_{2}mathbf{x}] +_2 [btimes_{2}mathbf{y}]$ , then we have the equation of a line that intersects the point $atimes_{2}mathbf{x}$ for all $ain{mathbb{R}}$ . I.e, we end up with a line for which each point on said line is a point on another line, with the equation of the line intersecting the first line at each point having the same gradient vector.