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Equivalence Classes and Partitions

If R is an equivalence relation on a set A, the equivalence class of an element a in A is [a] = {x in A : x R a}. Equivalence classes are non-empty, pairwise disjoint or identical, and together they partition A.

In one line: One class is one block of related elements; different names can point to the same block.

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Understand it

An equivalence class collects all elements that are related to a chosen representative. Different representatives can name the same class. The important idea is that an equivalence relation separates the set into blocks with no overlap between distinct blocks.

Example

For integers under the relation a R b if a - b is divisible by 3, the equivalence classes are [0], [1], and [2], representing integers that leave remainders 0, 1, and 2 on division by 3.

Formula and rule

Equivalence class: [a] = {x in A : x R a}.

Partition property: union of all distinct equivalence classes is A, and any two distinct classes have empty intersection.

If [a] and [b] have one common element, then [a] = [b].

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let A = {1,2,3,4,5,6} and define a R b if a and b have the same parity.

  1. 1Class [1] = {1,3,5} because these elements are odd like 1.
  2. 2Class [2] = {2,4,6} because these elements are even like 2.
  3. 3Now [3] = {1,3,5}, same as [1], so it is not a new class.
  4. 4Distinct classes are {1,3,5} and {2,4,6}; their union is A and their intersection is empty.

Method you can reuse

  1. 1First confirm or use the given fact that R is an equivalence relation.
  2. 2Choose an element a and write the condition x R a.
  3. 3Solve the condition to list or describe all x in the class.
  4. 4Avoid repeating a class already obtained.
  5. 5Verify partition by checking union and no overlap between distinct classes.

Visual tip

A partition diagram with disjoint boxes can support the idea, but exam answers mainly require set notation and a clear representative-based description.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

For the relation on integers defined by a R b if a - b is divisible by 4, what is [1]?

Think first, then show answer
[1] is the set of all integers of the form 4k + 1, where k is an integer.

Common wrong answer

Writing [1] = {1} for a congruence-type relation, ignoring other elements that satisfy the same remainder or property.

Don't mix this up

Students often write too many classes by treating every element as a new class, even when two elements are already in the same class.

Exam tip

After finding a class, check whether a new representative is already included in an earlier class. If yes, it does not create a new class.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions ask students to form equivalence classes for a given relation, show that the classes partition the set, or identify when two representatives give the same class.
  1. 1

    Enough for 1 mark

    If R is an equivalence relation on a set A, the equivalence class of an element a in A is [a] = {x in A : x R a}. Equivalence classes are non-empty, pairwise disjoint or identical, and together they partition A.

  2. 2

    Add for 2 marks

    Example: For integers under the relation a R b if a - b is divisible by 3, the equivalence classes are [0], [1], and [2], representing integers that leave remainders 0, 1, and 2 on division by 3.

  3. 3

    Add for 3 marks

    An equivalence class collects all elements that are related to a chosen representative.

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