Q.Let $R$ be the relation on $\mathbb{Z}$ defined by $R = \{(a, b) : a, b \in \mathbb{Z},\ a - b\text{ is an integer}\}$. Find the domain and range of $R$.

Relation Arrow Diagram

Imagine two groups of people at a party: one group of hosts, one group of guests. If you drew a string from every host to every guest they personally invited, you'd get a tangle of strings connecting the two groups. A relation arrow diagram does exactly this for sets — it draws an arrow from each element that is "related to" an element in the other set, so you can see a relation instead of just listing ordered pairs.

The Intuition First

A relation pairs up elements of one set with elements of another. Suppose:

• Set $A = \{\text{Ravi}, \text{Anu}, \text{Kiran}\}$ (students)
• Set $B = \{\text{Maths}, \text{Physics}, \text{Chemistry}\}$ (subjects)

and the relation is "studies": Ravi studies Maths and Physics; Anu studies only Chemistry; Kiran studies all three.

To draw the arrow diagram, list the elements of $A$ in one oval on the left, the elements of $B$ in another oval on the right, and draw one arrow for every pairing:

• Ravi $\rightarrow$ Maths
• Ravi $\rightarrow$ Physics
• Anu $\rightarrow$ Chemistry
• Kiran $\rightarrow$ Maths
• Kiran $\rightarrow$ Physics
• Kiran $\rightarrow$ Chemistry

Count the arrows leaving each name in $A$: Ravi has 2, Anu has 1, Kiran has 3 — six arrows in total, one for each ordered pair in the relation. That is the whole diagram: two ovals of labelled points, joined by one arrow per related pair.

The Precise Definition

Every arrow corresponds to exactly one ordered pair in $R$. If a diagram has $k$ arrows, the relation has exactly $k$ ordered pairs — nothing more, nothing less.

Key Points to Remember

• Direction matters. An arrow always starts at an element of $A$ and ends at an element of $B$. An arrow "Ravi $\to$ Maths" is not the same statement as "Maths $\to$ Ravi" — the first element of the ordered pair is always where the arrow starts.
• An element of $A$ can send multiple arrows. Kiran, above, sends three — one element can be related to many elements of $B$.
• An element of $A$ can send zero arrows. Nothing requires every element of $A$ to be related to something in $B$. If a student studies none of the listed subjects, no arrow leaves their name.
• An element of $B$ can receive multiple arrows. Both Ravi and Kiran point to Maths — an element of $B$ can be related to many elements of $A$.

A Relation on a Single Set

When a relation goes from a set to itself, both ovals contain the same elements, so they are usually drawn as one oval with arrows looping back into it. …