If R is an equivalence relation on a set A, which statement about its equivalence classes is correct?
Answer and explanation
Correct answer: They form a partition of A
For an equivalence relation R, the equivalence class of an element a is [a] = {x in A : xRa}. Reflexivity ensures that a belongs to [a], so every class is non-empty and all elements of A are covered. If two classes have even one common element, symmetry and transitivity show that their representatives are related, so the two classes are actually identical. Thus distinct classes are disjoint, while their union is A. These are precisely the defining features of a partition of A. Therefore option A is correct. Classes need not be singletons; for example, congruence modulo 2 gives classes of several elements. They are not empty, and distinct classes do not overlap.
Frequently asked questions
What is the correct answer to this question?
They form a partition of A
Why is this the correct answer?
For an equivalence relation R, the equivalence class of an element a is [a] = {x in A : xRa}. Reflexivity ensures that a belongs to [a], so every class is non-empty and all elements of A are covered. If two classes have even one common element, symmetry and transitivity show that their representatives are related, so the two classes are actually identical. Thus distinct classes are disjoint, while their union is A. These are precisely the defining features of a partition of A. Therefore option A is correct. Classes need not be singletons; for example, congruence modulo 2 gives classes of several elements. They are not empty, and distinct classes do not overlap.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Equivalence relation.
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