Let `A={1,2,3ddot, 14}` . Define a relation on a set A by `R={(x , y):3x-y=0. w h e r e\ x , y in A}` . Depict this relationship using an arrow diagram. Write down its domain, co-domain and range.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the Set A
We start with the set \( A = \{1, 2, 3, \ldots, 14\} \).

### Step 2: Define the Relation R
The relation \( R \) is defined by the equation \( 3x - y = 0 \) or equivalently \( y = 3x \). Here, \( x \) and \( y \) are elements of set \( A \).

### Step 3: Find Ordered Pairs in R
We will find the ordered pairs \( (x, y) \) such that both \( x \) and \( y \) belong to set \( A \).

- For \( x = 1 \): \( y = 3(1) = 3 \) → Pair: \( (1, 3) \)
- For \( x = 2 \): \( y = 3(2) = 6 \) → Pair: \( (2, 6) \)
- For \( x = 3 \): \( y = 3(3) = 9 \) → Pair: \( (3, 9) \)
- For \( x = 4 \): \( y = 3(4) = 12 \) → Pair: \( (4, 12) \)
- For \( x = 5 \): \( y = 3(5) = 15 \) → 15 is not in A, so we stop here.
- Continuing with \( x = 6, 7, \ldots, 14 \) will yield \( y \) values greater than 15, which are also not in A.

Thus, the relation \( R \) consists of the pairs:
\[ R = \{(1, 3), (2, 6), (3, 9), (4, 12)\} \]

### Step 4: Draw the Arrow Diagram
In the arrow diagram, we will represent the elements of set \( A \) on one side and their corresponding \( y \) values on the other side:

```
1 → 3
2 → 6
3 → 9
4 → 12
```

### Step 5: Determine Domain, Co-domain, and Range
- **Domain**: The set of all \( x \) values that have corresponding \( y \) values in the relation. From our pairs, the domain is:
\[ \text{Domain} = \{1, 2, 3, 4\} \]

- **Co-domain**: The set from which \( y \) values are taken. Since \( y \) values are derived from set \( A \), the co-domain is:
\[ \text{Co-domain} = A = \{1, 2, 3, \ldots, 14\} \]

- **Range**: The set of all \( y \) values that are actually produced by the relation. From our pairs, the range is:
\[ \text{Range} = \{3, 6, 9, 12\} \]

### Summary of Results
- **Relation R**: \( \{(1, 3), (2, 6), (3, 9), (4, 12)\} \)
- **Domain**: \( \{1, 2, 3, 4\} \)
- **Co-domain**: \( \{1, 2, 3, \ldots, 14\} \)
- **Range**: \( \{3, 6, 9, 12\} \)