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Let S = 1, 2, 3, ..., 10. Suppose M is the set of all the subsets of S, then the relation R = (A, B) : A ∩ B ≠ ∅; A, B ∈ M is:

Concept: Properties of relations

Question

Let S = {1, 2, 3, ..., 10}. Suppose M is the set of all the subsets of S, then the relation R = {(A, B) : A ∩ B ≠ ∅; A, B ∈ M} is:
A) Symmetric and reflexive only
B) reflexive only
C) symmetric and transitive only
D) symmetric only

Try it yourself first

Hint: To check reflexivity, consider if every set A in M must intersect with itself. What happens if A is the empty set?

Step-by-step solution

  1. 1

    Check Reflexivity

    Reflexive: (A,A) ∈ R? • A ∩ A ≠ ∅?
    A∩A≠∅A \cap A \neq \emptyset
  2. 2

    Reflexivity Test

    If A = ∅, A ∩ A = ∅ • Not reflexive
  3. 3

    Check Symmetry

    Symmetric: (A,B) ∈ R ⇒ (B,A) ∈ R? • A ∩ B ≠ ∅ ⇒ B ∩ A ≠ ∅?
    A∩B≠∅  ⟹  B∩A≠∅A \cap B \neq \emptyset \implies B \cap A \neq \emptyset
  4. 4

    Symmetry Test

    A ∩ B = B ∩ A • Is symmetric
  5. 5

    Check Transitivity

    Transitive: (A,B) ∈ R, (B,C) ∈ R ⇒ (A,C) ∈ R? • A ∩ B ≠ ∅, B ∩ C ≠ ∅ ⇒ A ∩ C ≠ ∅?
    (A∩B≠∅∧B∩C≠∅)  ⟹  A∩C≠∅(A \cap B \neq \emptyset \land B \cap C \neq \emptyset) \implies A \cap C \neq \emptyset
  6. 6

    Transitivity Test

    Counterexample: A={1,2}, B={2,3}, C={3,4} • A ∩ B ≠ ∅, B ∩ C ≠ ∅ • A ∩ C = ∅

Final answer

D) symmetric only

Remember: Always check all three properties: reflexive, symmetric, transitive.

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Quick check · test yourself

Let X = {a, b, c}. Consider the power set P(X). A relation R on P(X) is defined as (A, B) ∈ R if A is a proper subset of B (A⊂BA \subset B). Is R reflexive?

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