Let X be a non empty set and S be the collection of all subsets of X. A relation R in S defined by R = {(A, B): A⊂B}, where X⊂Y means X is proper subset of Y. Then R is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Transitive only
Here \( A\subset B \) means A is a proper subset of B, so \( A\ne B \).
Not reflexive: no set is a proper subset of itself, so \( (A,A)\notin R \); for example \( (X,X)\notin R \).
Not symmetric: the empty set φ is a proper subset of X (X is non-empty), so \( (\phi,X)\in R \); but X is not a proper subset of φ, so \( (X,\phi)\notin R \).
Transitive: if \( A\subset B \) and \( B\subset C \), then every element of A is in C, and C contains an element of B not in A, so \( A\subset C \). Hence \( (A,B),(B,C)\in R\Rightarrow(A,C)\in R \).
So R is transitive only.
Tip: The word “proper” is what breaks reflexivity: \( \subseteq \) would be reflexive and transitive, while \( \subset \) is only transitive.