The phrase “the graph of the relation s is shown below” appears frequently in textbooks, worksheets, and online exercises, signaling that a visual representation of a binary relation named S follows. This sentence serves as a bridge between abstract algebraic definitions and concrete pictorial understanding, allowing students to translate set‑theoretic concepts into a diagram that can be analyzed visually.
What Does the Phrase Mean?
When an author writes “the graph of the relation s is shown below”, they are referring to a specific mathematical object: a set of ordered pairs that together constitute the relation S. The word graph here does not refer to a function plot on Cartesian axes; instead, it denotes a directed graph (or digraph) in which each ordered pair ((a, b)) is represented by an arrow from vertex (a) to vertex (b). Recognizing this distinction is crucial for interpreting the diagram correctly.
Breaking Down the Sentence
- “the graph” – the visual diagram consisting of points (vertices) and connecting arrows (edges).
- “of the relation” – indicates that the diagram encodes a binary relation, a subset of the Cartesian product of two sets.
- “s” – the name of the relation, often capitalized as S; it may be defined elsewhere in the text.
- “is shown below” – directs the reader to examine the accompanying illustration.
Understanding each component helps prevent the common mistake of treating the graph as a standard x‑y plot.
Components of a Graph of a Relation
A graph of a relation is built from two fundamental elements:
- Vertices (or nodes) – each vertex corresponds to an element of the underlying set(s). If the relation is defined on a single set (A), the vertices are the members
