Relations — Definition, Pictorial Diagrams, Domain, Co-domain and Range

Relation. Let $A$ and $B$ be two non-empty sets. A relation $R$ from $A$ to $B$ is any subset of the Cartesian product $A \times B$. If $(a, b) \in R$, we say $a$ is related to $b$ under $R$, and write $a\,R\,b$.

Since $A \times B$ itself has $n(A) \times n(B)$ elements when $A, B$ are finite, and a relation is any subset of it, the total number of possible relations from $A$ to $B$ is $2^{n(A)\times n(B)}$ (the number of subsets of a set with $n(A)\times n(B)$ elements) — including the empty relation $R = \varnothing$ and the universal relation $R = A \times B$ itself.

Ways to describe a relation. A relation can be given (i) as a set of ordered pairs (roster form), e.g. $R = \{(1,2), (2,4), (3,6)\}$; (ii) by a defining rule/condition on $A \times B$ (set-builder form), e.g. $R = \{(x,y) : x \in A, y \in B, y = 2x\}$; or (iii) pictorially, by an arrow diagram.

Arrow (pictorial) diagram. Sets $A$ and $B$ are each drawn as a closed curve with their elements marked as points inside. For every ordered pair $(a, b) \in R$, an arrow is drawn from the point $a$ in $A$ to the point $b$ in $B$. Unlike a function's diagram, an element of $A$ under a general relation may have no arrow leaving it, exactly one arrow, or more than one arrow — there is no restriction at all on how many times an element of $A$ can appear as a first component.

Domain, co-domain and range of a relation. Let $R$ be a relation from $A$ to $B$.

• The domain of $R$ is the set of all first components of the ordered pairs in $R$: $\text{Dom}(R) = \{a : (a,b) \in R \text{ for some } b \in B\}$. It is a subset of $A$ (not necessarily equal to $A$).
• The co-domain of $R$ is the set $B$ itself, fixed in advance as part of specifying the relation — it need not consist only of elements that are actually related to something. …