If `A={1,2,3}` and `B={1,3,4,7}` and R is a relation from A to B defined by x is greater than y', then the range of R is

Options

A. [1,3,4,7]
B. [3,4,7]
C. [1]
D. none of these

Answer

To find the range of the relation \( R \) defined by the condition "x is greater than y" where \( A = \{1, 2, 3\} \) and \( B = \{1, 3, 4, 7\} \), we will follow these steps:

### Step 1: Identify the elements of sets A and B
- Set \( A \) contains the elements \( 1, 2, 3 \).
- Set \( B \) contains the elements \( 1, 3, 4, 7 \).

### Step 2: Define the relation R
- The relation \( R \) consists of ordered pairs \( (x, y) \) such that \( x \in A \), \( y \in B \), and \( x > y \).

### Step 3: Compare elements of A with elements of B
- We will check each element of set \( A \) against each element of set \( B \) to see if the condition \( x > y \) holds true.

1. For \( x = 1 \):
- \( 1 > 1 \) (False)
- \( 1 > 3 \) (False)
- \( 1 > 4 \) (False)
- \( 1 > 7 \) (False)
- No valid pairs for \( x = 1 \).

2. For \( x = 2 \):
- \( 2 > 1 \) (True) → Pair \( (2, 1) \)
- \( 2 > 3 \) (False)
- \( 2 > 4 \) (False)
- \( 2 > 7 \) (False)
- Valid pair: \( (2, 1) \).

3. For \( x = 3 \):
- \( 3 > 1 \) (True) → Pair \( (3, 1) \)
- \( 3 > 3 \) (False)
- \( 3 > 4 \) (False)
- \( 3 > 7 \) (False)
- Valid pair: \( (3, 1) \).

### Step 4: List the valid pairs in relation R
- From the comparisons, we found the following valid pairs:
- \( (2, 1) \)
- \( (3, 1) \)

Thus, the relation \( R = \{(2, 1), (3, 1)\} \).

### Step 5: Determine the range of R
- The range of the relation \( R \) consists of the second elements (y-values) from the valid pairs.
- From the pairs \( (2, 1) \) and \( (3, 1) \), the second elements are both \( 1 \).

### Conclusion
- Therefore, the range of the relation \( R \) is \( \{1\} \).

### Final Answer
The range of \( R \) is \( \{1\} \).

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