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What are the smallest and largest equivalence relation on a set?

Posted on June 1, 2021 by Author

Table of Contents

  • 1 What are the smallest and largest equivalence relation on a set?
  • 2 What is the smallest equivalence relation on a ABC?
  • 3 What is the smallest equivalence relation on the set 456?
  • 4 Which is the smallest relation?
  • 5 How many equivalence relations are possible in a set a 1/2 3?
  • 6 How many elements are there in the smallest equivalence relation on a set with 8 elements?
  • 7 How do you determine equivalence relations?

What are the smallest and largest equivalence relation on a set?

Explanation: An Equivalence relation is always Reflexive, Symmetric and Transitive, so for a set of size ‘n’ elements the largest Equivalence relation will always contain n2 elements whereas the smallest Equivalence relation on a set of ‘n’ elements contain n elements itself.

What is the smallest equivalence relation on a ABC?

A relation is an equivalence relation if and only if it is reflexive, symmetric and transitive:

  • A relation is an equivalence relation if and only if it is reflexive, symmetric and transitive:
  • The smallest equivalence relation on the set A={1,2,3} is :
  • R={(1,1),(1,3),(3,1)}
  • ∵ (1,1) ∈ R → Reflexive.
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What are the cardinalities of the largest and smallest equivalence relations on S?

to find the smallest and largest number of equivalence relation in a set. Let s be a set of n elements. The number of ordered pairs in the largest and smallest equivalence relation on set s are n2 and n.

What is the smallest equivalence relation on the set 456?

Answer: The smallest equivalence relation on the set A={4,5,6} is R={(4,4),(5,5),(6,6)}.

Which is the smallest relation?

The smallest one will be the one that is a subset of all the others. Among reflexive relations, the one you give is the smallest because those three pairs must be in every reflexive relation. You can add any other pairs you wish without spoiling the fact that the relation is reflexive.

What is the total number of equivalence relations that can be defined on the set 1 2 3?

R5 = {(1,2,3)4=. AxA =A2} Maximum number of equivalence relation is ‘5’.

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How many equivalence relations are possible in a set a 1/2 3?

Hence, only two possible relations are there which are equivalence. Note- The concept of relation is used in relating two objects or quantities with each other.

How many elements are there in the smallest equivalence relation on a set with 8 elements?

The smallest equivalence relation means it should contain minimum number of ordered pairs i.e along with symmetric and transitive properties it must always satisfy reflexive property. So, the smallest equivalence relation will have n ordered pairs and so the answer is 8.

What is the maximum number of equivalence relations on the set a ABC?

An equivalence relation is one which is reflexive, symmetric and transitive. We can define equivalence relation on A as follows. ∴ maximum number of equivalence relation on A is ‘5’.

How do you determine equivalence relations?

To prove an equivalence relation, you must show reflexivity, symmetry, and transitivity, so using our example above, we can say:

  1. Reflexivity: Since a – a = 0 and 0 is an integer, this shows that (a, a) is in the relation; thus, proving R is reflexive.
  2. Symmetry: If a – b is an integer, then b – a is also an integer.
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