Let A = {1,2,3,4,5,6} and R be the relation defined on A by `R = {(x, y): x, y in A, x` divides y}, then range of R is

Options

A. {2,3,4,5,6}
B. (1,2,3,4,5)
C. {2,4,6}
D. {1,2,3,4,5,6}

Answer

To find the range of the relation \( R \) defined on the set \( A = \{1, 2, 3, 4, 5, 6\} \) where \( R = \{(x, y) : x, y \in A, x \text{ divides } y\} \), we will follow these steps:

### Step 1: Understand the Relation
The relation \( R \) consists of ordered pairs \( (x, y) \) such that \( x \) divides \( y \). This means that for each pair, \( y \) must be a multiple of \( x \).

### Step 2: Identify Possible Ordered Pairs
We will find all pairs \( (x, y) \) for \( x, y \in A \) such that \( x \) divides \( y \).

- For \( x = 1 \): \( 1 \) divides \( 1, 2, 3, 4, 5, 6 \) → pairs: \( (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) \)
- For \( x = 2 \): \( 2 \) divides \( 2, 4, 6 \) → pairs: \( (2, 2), (2, 4), (2, 6) \)
- For \( x = 3 \): \( 3 \) divides \( 3, 6 \) → pairs: \( (3, 3), (3, 6) \)
- For \( x = 4 \): \( 4 \) divides \( 4 \) → pair: \( (4, 4) \)
- For \( x = 5 \): \( 5 \) divides \( 5 \) → pair: \( (5, 5) \)
- For \( x = 6 \): \( 6 \) divides \( 6 \) → pair: \( (6, 6) \)

### Step 3: Compile All Ordered Pairs
From the above, we compile the ordered pairs:
- From \( x = 1 \): \( (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) \)
- From \( x = 2 \): \( (2, 2), (2, 4), (2, 6) \)
- From \( x = 3 \): \( (3, 3), (3, 6) \)
- From \( x = 4 \): \( (4, 4) \)
- From \( x = 5 \): \( (5, 5) \)
- From \( x = 6 \): \( (6, 6) \)

Thus, the complete set of ordered pairs in \( R \) is:
\[ R = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (5, 5), (6, 6)\} \]

### Step 4: Determine the Range
The range of a relation is the set of all second elements (the \( y \) values) from the ordered pairs.

From our ordered pairs, the second elements are:
- From \( (1, 1) \): \( 1 \)
- From \( (1, 2) \): \( 2 \)
- From \( (1, 3) \): \( 3 \)
- From \( (1, 4) \): \( 4 \)
- From \( (1, 5) \): \( 5 \)
- From \( (1, 6) \): \( 6 \)
- From \( (2, 2) \): \( 2 \)
- From \( (2, 4) \): \( 4 \)
- From \( (2, 6) \): \( 6 \)
- From \( (3, 3) \): \( 3 \)
- From \( (3, 6) \): \( 6 \)
- From \( (4, 4) \): \( 4 \)
- From \( (5, 5) \): \( 5 \)
- From \( (6, 6) \): \( 6 \)

Now, we combine these values without repetition:
\[ \text{Range of } R = \{1, 2, 3, 4, 5, 6\} \]

### Final Answer
The range of the relation \( R \) is:
\[ \{1, 2, 3, 4, 5, 6\} \]