Question

If \[ R=\{(x,y)\mid x,y\in \mathbb{R},\ x^2+y^2=1\} \] is a relation in \(\mathbb{R}\), then \(R\) is: 

• A. \( \{0,1,2\} \)
• B. \( \{0,-1,-2\} \)
• C. \( \{-2,-1,0,1,2\} \)
• D. \( \{-1,0,2\} \)

Hint:

To find the domain of a relation defined by an equation or inequality involving \(x\) and \(y\):
• Focus on the first coordinate \(x\).
• Find all values of \(x\) for which at least one valid value of \(y\) exists.
• For inequalities involving squares, use the fact that: \[ x^2\geq 0,\qquad y^2\geq 0 \] to simplify the range quickly. In integer-based relations, always check whether the required values are integers.

Answer 1

The Correct Option is C

Concept: A relation \(R\) from a set \(A\) to a set \(B\) is a collection of ordered pairs. The domain of a relation consists of all first elements of the ordered pairs belonging to the relation. If \[ R=\{(x,y)\mid \text{condition on }x\text{ and }y\} \] then the domain is the set of all values of \(x\) for which at least one value of \(y\) exists satisfying the given condition. In this problem: \[ R=\{(x,y)\mid x,y\in \mathbb{Z},\ x^2+y^2\leq 4\} \] Since both \(x\) and \(y\) are integers, we must find all integer values of \(x\) for which there exists at least one integer \(y\) satisfying: \[ x^2+y^2\leq 4 \]

Step 1: Understanding the given inequality We are given: \[ x^2+y^2\leq 4 \] Since squares are always non-negative, \[ x^2\leq 4 \] because \(y^2\geq 0\). Now solve: \[ x^2\leq 4 \] Taking square roots, \[ -2\leq x\leq 2 \] Since \(x\in \mathbb{Z}\), the possible integer values are: \[ x=-2,-1,0,1,2 \] Now we must check whether for each of these values there exists at least one integer value of \(y\) satisfying the inequality.

Step 2: Checking each possible value of \(x\) \(x=-2\) Substituting into the inequality: \[ (-2)^2+y^2\leq 4 \] \[ 4+y^2\leq 4 \] \[ y^2\leq 0 \] Thus, \[ y=0 \] which is an integer. Hence, \(x=-2\) belongs to the domain. \(x=-1\) \[ (-1)^2+y^2\leq 4 \] \[ 1+y^2\leq 4 \] \[ y^2\leq 3 \] Possible integer values: \[ y=-1,0,1 \] Hence, at least one integer \(y\) exists. Therefore, \(x=-1\) belongs to the domain. \(x=0\) \[ 0^2+y^2\leq 4 \] \[ y^2\leq 4 \] Possible integer values: \[ y=-2,-1,0,1,2 \] Hence, \(x=0\) belongs to the domain. \(x=1\) \[ 1+y^2\leq 4 \] \[ y^2\leq 3 \] Possible integer values: \[ y=-1,0,1 \] Thus, \(x=1\) belongs to the domain. \(x=2\) \[ 2^2+y^2\leq 4 \] \[ 4+y^2\leq 4 \] \[ y^2\leq 0 \] Hence, \[ y=0 \] Therefore, \(x=2\) also belongs to the domain.

Step 3: Writing the domain All possible values of \(x\) are: \[ \{-2,-1,0,1,2\} \] Therefore, the domain of the relation \(R\) is: \[ \boxed{\{-2,-1,0,1,2\}} \] Hence, the correct option is: \[ \boxed{(C)} \]