Q.Let $A = \{1, 2, 3, \ldots, 14\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : 3x - y = 0,\ \text{where}\ x, y \in A\}$. Write down its domain, codomain and range.

Concept: Relation Arrow Diagram — a relation $R$ from $A$ to $A$ is a set of ordered pairs $(x, y)$ satisfying a given condition. The domain is the set of all first elements, the range is the set of all second elements, and the codomain is the whole target set $A$.

Step 1: The condition $3x - y = 0$ means $y = 3x$. Since $x, y \in A = \{1, 2, \dots, 14\}$, we list all pairs where $3x \leq 14$.

Step 2: For $x = 1$, $y = 3$; $x = 2$, $y = 6$; $x = 3$, $y = 9$; $x = 4$, $y = 12$; $x = 5$, $y = 15$ (not in $A$). So only $x = 1, 2, 3, 4$ work.

Step 3:

• Domain = $\{1, 2, 3, 4\}$
• Codomain = $A = \{1, 2, 3, \dots, 14\}$
• Range = $\{3, 6, 9, 12\}$

The key idea is to see the relation as a mapping: each $x$ in $A$ is linked to exactly one $y$ in $A$ through the equation $y = 3x$. But not every $x$ in $A$ will produce a $y$ that stays inside $A$ — and that's what decides the domain and range.

Think of drawing an arrow diagram. You have set $A$ on the left and the same set $A$ on the right. For each $x$ in the left set, you check if $3x$ is a number in the right set. If yes, you draw an arrow from $x$ to $3x$. The domain is the set of all left-side numbers that have an arrow leaving them. The range is the set of all right-side numbers that have an arrow arriving at them. The codomain is simply the entire right-side set — whether or not any arrow reaches it.

Now let's work through it.

1. Rewrite the condition.

   The relation is $R = \{(x, y) : 3x - y = 0, x, y \in A\}$.

   This simplifies to $y = 3x$. So for each $x$, the corresponding $y$ is forced to be $3x$.

2. Find which $x$ values are allowed.

   Since $x$ must be in $A = \{1, 2, 3, \ldots, 14\}$, and $y = 3x$ must also be in $A$, we need $3x \leq 14$.

   Solve $3x \leq 14 \implies x \leq \frac{14}{3} \approx 4.67$.

   Since $x$ is a positive integer, the possible $x$ values are $1, 2, 3, 4$.

3. List the ordered pairs.

   For $x = 1$, $y = 3$ → $(1,3)$

   For $x = 2$, $y = 6$ → $(2,6)$

   For $x = 3$, $y = 9$ → $(3,9)$

   For $x = 4$, $y = 12$ → $(4,12)$

   For $x = 5$, $y = 15$ — but $15 \notin A$, so no pair.

   So $R = \{(1,3), (2,6), (3,9), (4,12)\}$.

4. Identify the domain.

   The domain is the set of all first elements in the ordered pairs.

   Domain $= \{1, 2, 3, 4\}$.

5. Identify the codomain.

   The codomain is given as the set to which $y$ belongs — here it's $A$ itself.

   Codomain $= \{1, 2, 3, \ldots, 14\}$.

6. Identify the range.

   The range is the set of all second elements that actually appear in the relation.

   Range $= \{3, 6, 9, 12\}$.