Q.Define a relation $R$ on the set $\mathbb{N}$ of natural numbers by $R = \{(x, y) : y = x + 5,\ x\ \text{is a natural number less than }4;\ x, y \in \mathbb{N}\}$. Depict this relationship using roster form. Write down the domain and the range.

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.

Example. Let $A = \{1, 2, 3\}$ and let $R$ be "is less than" ($<$). The pairs in $R$ are $(1,2), (1,3), (2,3)$, so the arrow diagram has exactly these three arrows:

• $1 \rightarrow 2$
• $1 \rightarrow 3$
• $2 \rightarrow 3$

Nothing points from $3$, since no element of $A$ is greater than $3$. Nothing points into $1$, since no element of $A$ is less than $1$.

Why This Matters

Arrow diagrams are the first visual bridge toward functions, one-one relations, onto relations, and equivalence relations. Once you can draw one, you can immediately see properties that are hard to spot from a bare list of ordered pairs — such as whether an element sends more than one arrow (which would rule the relation out as a function), or whether some element in $B$ is never hit by any arrow.

Representing a relation using an arrow diagram is a visual technique taught early in the NCERT Class 11 Mathematics chapter on Relations and Functions, and "relation arrow diagram examples class 11" is a frequently searched revision topic for CBSE board preparation. This visualisation also makes it easier to identify one-one and onto relations, a distinction that is regularly tested in "relations and functions important questions".