Define a relation on a set. What do you mean by the domain and range of a relation?

Question:

Define a relation on a set. What do you mean by the domain and range of a relation? Give an example.

 

Solution:

Relation: Let $A$ and $B$ be two sets. Then a relation $R$ from set $A$ to set $B$ is a subset of $A \times B$. Thus, $R$ is a relation to $A$ to $B \Leftrightarrow R \subseteq A \times B$.

If $R$ is a relation from a non-void set $B$ and if $(a, b) \in R$, then we write $a$ R b which is read as ' $a$ is related to $b$ by the relation $R^{\prime}$. if $(a, b) \notin R$, then we write $a$ R $b$, and we say that a is not related to $b$ by the relation $R$.

Domain: Let $R$ be a relation from a set $A$ to a set $B$. Then the set of all first components or coordinates of the ordered pairs belonging to $R$ is called the domain of $R$.

Thus, domain of $R=\{a:(a, b) \in R\} .$ The domain of $R \subseteq A .$

Range: let $R$ be a relation from a set $A$ to a set $B$. then the set of all second component or coordinates of the ordered pairs belonging to $R$ is called the range of $R$.

Example 1: $R=\{(-1,1),(1,1),(-2,4),(2,4)\}$

dom $(R)=\{-1,1,-2,2\}$ and range $(R)=\{1,4\}$

Example 2: $R=\{(a, b): a, b \in N$ and $a+3 b=12\}$

dom $(R)=\{3,6,9\}$ and range $(R)=\{3,2,1\}$

 

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