Questions: Binary Relations

Question 1

Let R be the relation on the integers defined by 'a R b iff a ≤ b.' Which properties does R have?

• A. Reflexive and symmetric only
• B. Symmetric and transitive only
• C. Reflexive and transitive only
• D. Reflexive, symmetric, and transitive

Answer: (C)

≤ is reflexive (a ≤ a), transitive (if a ≤ b and b ≤ c then a ≤ c), but NOT symmetric (2 ≤ 3 does not imply 3 ≤ 2). A relation that is reflexive and transitive but not symmetric is a preorder (or partial order if also antisymmetric).

Question 2

Every subset of A × A qualifies as a binary relation on A.

• T. True
• F. False

Answer: True

By definition, a binary relation on A is simply any subset of A × A. There is no requirement that the subset satisfy reflexivity, symmetry, or any other property — those are additional properties a relation may or may not have. The empty set and the full Cartesian product are both valid (if extreme) relations.

Question 3

Give an example of a relation that is symmetric and transitive but NOT reflexive, or explain why no such relation can exist on a non-empty set.

Model answer: The empty relation on any non-empty set A is symmetric and transitive (vacuously) but not reflexive. Among non-empty relations, consider R = {(1,2),(2,1),(1,1),(2,2)} on {1,2,3} — it is symmetric and transitive on the elements it relates, but not reflexive because (3,3) is absent.

It is a common misconception that a symmetric + transitive relation must be reflexive. The reasoning 'if a R b and b R a, then a R a by transitivity' only works if every element appears in some pair. Elements that appear in no pair are never forced to be self-related, so reflexivity can fail.