Quant Exam  >  Quant Questions  >  The set of all Equivalence classes of a set A... Start Learning for Free
The set of all Equivalence classes of a set A of cardinality C
  • a)
    has the same cardinality as A
  • b)
    forms a partition of A
  • c)
    is of cardinality 2C
  • d)
    is of cardinality C2
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The set of all Equivalence classes of a set A of cardinality Ca)has th...
The equivalence classes of any equivalence relation on A partition A. There's no need to talk about cardinalities to know this; that's just the fact that equivalence relations are equivalent to partitions.

If A is the natural numbers, and we take just one equivalence class (all of A), then a), b), d) claim that there are infinitely many equivalence classes. But there's just one.
Free Test
Community Answer
The set of all Equivalence classes of a set A of cardinality Ca)has th...
Equivalence Classes and Cardinality

Equivalence classes are sets that contain elements that are equivalent to each other based on a certain relation. For example, if we consider the relation of congruence modulo 5, then the equivalence class of 2 would contain all integers that leave a remainder of 2 when divided by 5, such as {2, 7, 12, -3, ...}.

Cardinality is a measure of the size of a set, which is given by the number of elements in the set. For example, the cardinality of the set {1, 2, 3, 4, 5} is 5.

Equivalence Classes of a Set A

If we consider the set A and a certain relation R on A, then the equivalence classes of A under R form a partition of A. This means that every element of A belongs to exactly one equivalence class, and two elements belong to the same equivalence class if and only if they are equivalent under R.

For example, if we consider the set A = {1, 2, 3, 4, 5} and the relation R of congruence modulo 2, then the equivalence classes of A under R would be {1, 3, 5} and {2, 4}.

Cardinality of Equivalence Classes

The cardinality of an equivalence class is the number of elements in the equivalence class. If we denote the cardinality of an equivalence class [a] by |[a]|, then we have:

|[a]| = |{x ∈ A | xRa}|,

where a is an element of A, and Ra denotes the equivalence relation between a and x.

If we consider all the equivalence classes of A under R, then these equivalence classes will form a partition of A. This means that every element of A belongs to exactly one equivalence class, and the union of all the equivalence classes is equal to A.

Therefore, we have:

|A| = ∑ |[a]|,

where the sum is taken over all equivalence classes of A under R.

Since every element of A belongs to exactly one equivalence class, we can also write:

|A| = ∑ |[a]| = ∑ |[x]|,

where the sum is taken over all elements x of A.

This means that the set of all equivalence classes of A has the same cardinality as A.
Explore Courses for Quant exam

Similar Quant Doubts

The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer?
Question Description
The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer? for Quant 2025 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Quant 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer?.
Solutions for The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Quant. Download more important topics, notes, lectures and mock test series for Quant Exam by signing up for free.
Here you can find the meaning of The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer?, a detailed solution for The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice The set of all Equivalence classes of a set A of cardinality Ca)has the same cardinality as Ab)forms a partition of Ac)is of cardinality 2Cd)is of cardinality C2Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Quant tests.
Explore Courses for Quant exam
Signup to solve all Doubts
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev