A Relation - mathSight

A relationship can be viewed as a statement about an ordered that is true. We can make this statement as ${R(a,b)}$ where ${R}$ is the relation on the ordered pair. If we have the set of all ordered pairs ${A \times B}$ then the set of these pairs that satisfy ${R(a,b)}$ are a subset of ${A \times B}$ .

$\displaystyle \begin{array}{rcl} \forall a\in A \forall b\in B [ R(a,b) \subset A \times B] \end{array}$

Definition: Let ${R}$ be a relation and ${A}$ be a set. ${\forall x \in A (x,x)\in R}$ implies the relationship is reflexive.

Definition: Let ${R}$ be a relation and ${A}$ be a set. ${\forall x \in A (x,y)\in R \implies (y,x)\in R}$ .The relationship is symmetric.

Definition: Let ${R}$ be a relation and ${A}$ be a set. ${\forall x \in A (x,y)\in R \land (y,x)\in R \implies x=y}$ .The relationship is antisymmetric.

Definition: Let ${R}$ be a relation and ${A}$ be a set. ${\forall x \in A (x,y)\in R \land (y,z)\in R \implies (x,z)\in R}$ .The relationship is transitive.

If a relation is reflexive, symmetric and transitive then it is called an equivalence relation. If a relation is reflexive, antisymmetric and transitive then it is an order relation.

We now have the elements to define a very common term in mathematics, a function, much more precisely using set theory. A function is a binary (meaning it takes two elements) relation ${f}$ . We can state this as ${(x,y)\in f}$ . To be sure, this states that ${f}$ relates ${x}$ to ${y}$ . However, to be a function we need to make sure that ${x}$ can only relate to one element. This means if ${(x,y)\in f}$ and ${(x,z)\in f}$ then ${y=z}$ . The unique ${y}$ is called the value of ${f}$ at ${x}$ . This is commonly written ${f(x) = y}$ .

From the definition of a set we have derived a definition for an ordered pair which leads to the concept of a cartesian product. A relation is a true statement about a subsection of a cartesian product. We can define a special case which we call a function. We make a final observation that follows from this.

$\begin{equation*} \forall x\in X \; \forall y\in Y \; (x,y)\in f \implies f \subseteq X \times Y \end{equation*}$

That is, a function is a relation that is a subset of a cartesian product of sets.

A.D. (David) Everett : “Large streams from little fountains flow,
Tall oaks from little acorns grow.”