Question

Given \(A = \{1,2,3,4,5\}\), \(B = \{1,4,5\}\). If \(R\) is a relation from \(A\) to \(B\) such that \((x,y) \in R\) with \(x>y\), then the range of \(R\) is

• A. \( \{1,4\} \)
• B. \( \{4,5\} \)
• C. \( \{1,4,5\} \)
• D. \( \{2,4\} \)

Hint:

To find the range of a relation, always check which elements of the second set actually appear as second components in valid ordered pairs.

Answer 1

The Correct Option is A

Step 1: Understand the relation.
The relation \(R\) consists of all ordered pairs \((x,y)\) such that \[ x \in A,\; y \in B \quad \text{and} \quad x>y \] Step 2: List possible pairs.
For \(y = 1\): all \(x \in A\) such that \(x>1\) give valid pairs, so \(y = 1\) is included.
For \(y = 4\): values of \(x \in A\) satisfying \(x>4\) exist (namely \(x = 5\)), so \(y = 4\) is included.
For \(y = 5\): there is no \(x \in A\) such that \(x>5\), so \(y = 5\) is excluded.
Step 3: Determine the range.
The range of a relation consists of all second elements of the ordered pairs. Hence, \[ \text{Range}(R) = \{1,4\} \]