Questions: Binary Relations and Their Properties

Question 1

The relation R on the integers defined by 'aRb if a ≤ b' satisfies which combination of properties?

• A. Reflexive, symmetric, and transitive — it is an equivalence relation
• B. Reflexive, antisymmetric, and transitive — it is a partial (and total) order
• C. Symmetric and transitive but not reflexive
• D. Antisymmetric only — it fails both reflexivity and transitivity

Answer: (B)

The ≤ relation is reflexive (a ≤ a for all integers), antisymmetric (if a ≤ b and b ≤ a then a = b), and transitive (if a ≤ b and b ≤ c then a ≤ c). It is NOT symmetric: 3 ≤ 5 but 5 ≤ 3 is false. This combination — reflexivity, antisymmetry, transitivity — defines a partial order. Since every pair of integers is comparable (either a ≤ b or b ≤ a), it is additionally a total order.

Question 2

Consider the 'is a sibling of' relation. A student claims it is transitive because if Alice is a sibling of Bob and Bob is a sibling of Carol, then Alice is a sibling of Carol. Is 'sibling of' an equivalence relation?

• A. Yes — it is reflexive, symmetric, and transitive, satisfying all three requirements
• B. No — sibling-of is symmetric and transitive but fails reflexivity (no one is their own sibling), so it is not an equivalence relation
• C. No — sibling-of is not transitive, so the student's reasoning is flawed
• D. Yes — any relation that is symmetric and transitive is automatically reflexive

Answer: (B)

The student's transitivity claim is correct for full siblings. And sibling-of is symmetric. But reflexivity fails: a person is not their own sibling, so for person a, aRa does not hold. An equivalence relation requires all three properties: reflexivity, symmetry, and transitivity. Failing any one disqualifies it. Note that option D is a common misconception: a relation can be symmetric and transitive without being reflexive (if no element is related to anything, it vacuously satisfies both while failing reflexivity).

Question 3

Every equivalence relation on a set S partitions S into disjoint subsets called equivalence classes, where every element in a subset is related to every other element in that subset.

• T. True
• F. False

Answer: True

This is one of the most important theorems in basic set theory. An equivalence relation's three properties — reflexivity (every element is in some class), symmetry (membership is mutual), and transitivity (being related to the same element puts you in the same class) — together guarantee that the equivalence classes are disjoint and exhaustive. The integers modulo n provide a canonical example: congruence mod 3 partitions the integers into exactly three classes.

Question 4

A symmetric relation and an antisymmetric relation are mutually exclusive — no relation can satisfy both properties simultaneously.

• T. True
• F. False

Answer: False

False. The equality relation = on any set satisfies both. Symmetry requires: if aRb then bRa. Antisymmetry requires: if aRb and bRa then a = b. The equality relation satisfies symmetry (if a = b then b = a) and antisymmetry (if a = b and b = a then trivially a = b). There is no contradiction because antisymmetry's condition only triggers when both aRb and bRa hold — equality allows this only when a and b are the same element. The two properties conflict only when a relation has pairs (a,b) and (b,a) with a ≠ b.

Question 5

Explain the difference between an equivalence relation and a partial order in terms of the properties they require. Why does swapping symmetry for antisymmetry change the mathematical structure so fundamentally?

Model answer: Both equivalence relations and partial orders require reflexivity and transitivity. The difference is in the third property: equivalence relations add symmetry (if aRb then bRa), while partial orders add antisymmetry (if aRb and bRa then a = b). Symmetry means the relation is mutual — being related has no directionality, so all related elements form clusters of 'equal' elements. Antisymmetry means the relation has direction — when it runs both ways, the two elements must be identical. This turns clusters into a ranked structure where elements can be 'above' or 'below' each other. Equality becomes the only way to be related in both directions in an order, giving the structure its hierarchical character.

The three-property framework reveals how small changes in axioms produce radically different mathematical structures. Equivalence relations partition sets; partial orders impose hierarchy. The swap of one property — symmetry vs. antisymmetry — is the precise formal difference between 'same kind of thing' and 'at most as large as.' This is why identifying which properties a relation has is the first step to understanding what kind of mathematical structure it defines.