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IAT IISER2026MathematicsSets and Relations

Let C be the set of all the circles in a plane. If R = (C₁, C₂) C C C₁ and C₂ intersect , then which of the following statements is TRUE?

Options

  1. AR is reflexive and transitive but not symmetric.
  2. BR is not a relation.
  3. CR is symmetric and transitive but not reflexive.
  4. DR is reflexive and symmetric but not transitive.

Correct answer

D. R is reflexive and symmetric but not transitive.

Step-by-step solution

The relation R is defined as (C₁, C₂) R if C₁ and C₂ intersect. Reflexivity: For any circle C , the intersection of C with itself is C , which is non-empty ( C C = C ). Thus, C intersects itself, meaning (C, C) R . Therefore, R is reflexive. Symmetry: If C₁ intersects C₂ , then C₁ C₂ . This implies C₂ C₁ , so C₂ intersects C₁ . Thus, (C₁, C₂) R (C₂, C₁) R . Therefore, R is symmetric. Transitivity: Consider three circles C₁, C₂, C₃ of radius 1 centered at (0,0) , (2,0) , and (4,0) respectively. C₁ and C₂ intersect a

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