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NCERT Class 12 · Ch 1 Miscellaneous Exercise · Q3

NCERT Class 12 Maths · Chapter 1 Relations And Functions · Miscellaneous Exercise Question 3

For a nonempty set X, on P(X) define ARB iff A⊂B. Is this an equivalence relation? State the interpretation of ⊂.

💡 Concept Hints

  1. Use the empty set and a singleton to test symmetry.
  2. Write the defining condition carefully and preserve every domain restriction while simplifying.

✅ Step-by-Step Solution

Final Answer: No; not symmetric. The conclusion is unchanged if ⊂ is read as strict inclusion.
  1. Choose x∈X. Then ∅ is a subset of {x}, but {x} is not a subset of ∅, so symmetry fails.
  2. If ⊂ denotes inclusion allowing equality, the relation is reflexive and transitive, but the counterexample disproves equivalence.
  3. If ⊂ denotes proper inclusion, it is transitive but also fails reflexivity. In either convention it is not an equivalence relation.

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