Relations, Domain, and Range

Definition of a Relation

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product $A \times B$. This subset is created by defining a relationship between the first element and the second element of the ordered pairs in $A \times B$.

If an element 'a' from set A is related to an element 'b' from set B, we write $(a, b) \in R$. Here, 'b' is called the image of 'a'.

Domain, Range, and Codomain

A relation has three important components:

• Domain: The set of all first elements of the ordered pairs in a relation R is called the domain.
• Range: The set of all second elements (images) of the ordered pairs in a relation R is called the range.
• Codomain: The entire set B is called the codomain of the relation R from A to B.
• An important point is that the Range is always a subset of the Codomain (Range $\subseteq$ Codomain).

Total Number of Relations

The total number of relations that can be defined from a set A to a set B is the number of possible subsets of $A \times B$. If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = pq$. The total number of relations is $2^{pq}$.

Example 1: Finding Domain and Range

Question: Let $A = \{1, 2, 3, 4, 5, 6\}$. Define a relation R from A to A by $R = \{(x, y) : y = x + 1\}$. Write down the domain, codomain, and range of R.

Solution: By the definition, we find the pairs where the second element is one more than the first: $$R = \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)\}$$ (Note: $(6, 7)$ is not included because 7 is not in set A).

• Domain: The set of all first elements is $\{1, 2, 3, 4, 5\}$.
• Range: The set of all second elements is $\{2, 3, 4, 5, 6\}$.
• Codomain: The entire set A is the codomain, which is $\{1, 2, 3, 4, 5, 6\}$.