The maximum number of equivalence relations on the set $A = \{1$, $2$, $3\}$ are
- $1$
- $2$
- $3$
- $5$
The Correct Option is D
Approach Solution - 1
The smallest equivalence relation is the identity relation \(R_1 = \{(1\), \(1)\), \((2\), \(2)\), \((3\), \(3)\}\) Then, two ordered pairs of two distinct elements can be added to give three more equivalence relations. \(R_2 = \{(1\), \(1)\), \((2\), \(2)\), \((3\), \(3)\), \((1\), \(2)\), \((2\), \(1)\}\) Similarly \(R_3\) and \(R_4\). Finally the largest equivalence relation, that is the universal relation. \(R_5 = \{(1\), \(1)\), \((2\), \(2)\), \((3\), \(3)\), \((1\), \(2)\), \((2\), \(1)\), \((1\), \(3)\), \((3\), \(1)\), \((2\), \(3)\), \((3\), \(2)\}\)
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Approach Solution -2
The correct answer is Option D) 5
Real Life Applications
- Siblings: Two persons sharing at least one parent is called a sibling. It can be reflexive transitive and symmetric
- Classmates can be considered as an equivalence relation. Person 1 will be the classmate of person 2 and vice versa.
- People of the same religion or person speaking the same language
- People with the same blood type. If person 1 has the same blood group as person 2 if this person 2 has the same blood group with other people. Then person 1 will have the same blood as that of those people
Question can also be asked as
- How many equivalence relations are there on the set {1, 2, 3}?
- What is the maximum number of equivalence relations on a set with 3 elements?
Approach Solution -3
The correct answer is Option D) 5
The link between the two entities can be obtained by relation and functions. It maps the elements of domains with co-domains.
Relations: The subset of Cartesian products is called relation. It is the collection of ordered pair, the pair formed by objects taken from both sets.
Types of relations are as follows:
- Empty relation
- Universal relation
- Inverse relation
- Identity relation
- Reflexive relation
- Transitive relation
- Symmetric relation
Functions: It suggests that each given input should have a single output. In this, it is divided into domain and range.
The types of functions are as follows:
- Many to-one function
- Injective function/one-to-one function
- Bijective function
- Onto function
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Notes on types of relations
Concepts Used:
Types of Relation
TYPES OF RELATION
Empty Relation
Relation is said to be empty relation if no element of set X is related or mapped to any element of X i.e, R = Φ.
Universal Relation
A relation R in a set, say A is a universal relation if each element of A is related to every element of A.
R = A × A.
Identity Relation
Every element of set A is related to itself only then the relation is identity relation.
Inverse Relation
Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1
Reflexive Relation
If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.
Symmetric Relation
A relation R is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.
Transitive Relation
A relation is said to be transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A
Equivalence Relation
A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.