The relation R on integers defined by 'a R b if and only if a − b is even' — which of the following correctly describes its properties?
AReflexive and symmetric, but not transitive
BReflexive and transitive, but not symmetric
CReflexive, symmetric, and transitive — it is an equivalence relation
DSymmetric and transitive, but not reflexive
Reflexivity: a − a = 0, which is even. Symmetry: if a − b is even then b − a = −(a − b) is also even. Transitivity: if a − b and b − c are both even, their sum (a − c) = (a − b) + (b − c) is even. All three properties hold, making R an equivalence relation. Its two equivalence classes are the even integers and the odd integers.
Question 2 True / False
The relation 'less than' (<) on the real numbers is an equivalence relation because it is transitive.
TTrue
FFalse
Answer: False
Transitivity alone is not sufficient for an equivalence relation. 'Less than' fails reflexivity (no number satisfies a < a) and fails symmetry (a < b does not imply b < a). An equivalence relation requires all three properties: reflexive, symmetric, and transitive. Transitivity alone produces a strict partial order, not an equivalence relation.
Question 3 Short Answer
What is an equivalence class, and how do equivalence classes relate to the partition of a set?
Think about your answer, then reveal below.
Model answer: The equivalence class [a] is the set of all elements related to a. The equivalence classes of an equivalence relation partition the domain into non-overlapping, exhaustive subsets — every element belongs to exactly one class.
Reflexivity ensures a is always in its own class. Symmetry and transitivity together ensure two equivalence classes are either identical or completely disjoint — no element can belong to two different classes at once. So the classes tile the domain without gaps or overlaps, which is precisely the definition of a partition.