Relation, Domain, Co-domain and Range

The everyday word 'relation' — two objects or quantities are 'related' if there is a recognisable link between them (A is a friend of B, F is the father of S, P is the sister of Q) — is formalised in mathematics as follows. Among integers, for instance, we might define a relation R by 'm is a factor of n', writing $2R4$, $3R6$, $5R10$.

Formally, a well-defined relation between elements of A and B is written $aRb$ for $a\in A, b\in B$, and this relation gives rise to the ordered pair $(a,b)$ — so the relation defines a SUBSET of $A\times B$. Conversely, every subset of $A\times B$ defines a unique relation. So a RELATION FROM A TO B IS, BY DEFINITION, ANY SUBSET OF $A\times B$. For example, if $A=\{2,3,4,5,6\}$ and $B=\{6,7,8,10\}$, the relation 'a is a factor of b' gives the subset $\{(2,6),(2,8),(2,10),(3,6),(4,8),(5,10)\}$ of $A\times B$.

When $A=B$ (a relation among the elements of a single set A), it is a subset of $A\times A$, and is called a BINARY RELATION on A (covered further in the next section).

WORKED EXAMPLE: let $A=\{1,2,3,4,5\}$ and $B=\{1,4,5\}$, and let R be the relation where $(x,y)\in R$ means $x<y$. Listing all such pairs: $R=\{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)\}$ (Fig. 5.19 shows this as an arrow diagram from A to B).

DOMAIN: the set of all FIRST components of the ordered pairs in a relation R is called the domain of R: $\text{domain}(R) = \{a/(a,b)\in R\}$.

RANGE: the set of all SECOND components of the ordered pairs in R is called the range of R: $\text{range}(R) = \{b/(a,b)\in R\}$. …