Let S be the set of all persons living in Delhi. We say that x, y in S are related if they were born in Delhi on the same day. Which one of the following is correct?
The relation is an equivalence relation
Understanding Relations and Their Properties
The question asks us to analyze a specific relation defined on a set of people living in Delhi. We need to determine if this relation possesses certain properties: reflexivity, symmetry, and transitivity, and based on these properties, classify the type of relation.
Defining the Set and the Relation
Let S be the set of all persons living in Delhi.
A relation R is defined on S such that for any two persons x and y in S, x R y if and only if x and y were born in Delhi on the same day.
Analyzing the Properties of the Relation
To determine the type of relation, we check if it is reflexive, symmetric, and transitive.
1. Reflexivity of the Relation
A relation R on a set S is reflexive if for every element x ∈ S, (x, x) ∈ R. In simpler terms, an element must be related to itself.
- Consider any person x living in Delhi (x ∈ S).
- According to the relation definition, x R x if x was born in Delhi on the same day as x.
- Logically, a person is always born on the same day as themselves.
- Therefore, for any x ∈ S, x R x holds.
The relation is reflexive.
2. Symmetry of the Relation
A relation R on a set S is symmetric if for every x, y ∈ S, whenever (x, y) ∈ R, it implies that (y, x) ∈ R. In simpler terms, if x is related to y, then y must be related to x.
- Assume x R y for some x, y ∈ S.
- By definition, x R y means that x and y were born in Delhi on the same day.
- If x and y were born on the same day, it naturally follows that y and x were also born on the same day.
- Therefore, if x R y holds, then y R x also holds for any x, y ∈ S.
The relation is symmetric.
3. Transitivity of the Relation
A relation R on a set S is transitive if for every x, y, z ∈ S, whenever (x, y) ∈ R and (y, z) ∈ R, it implies that (x, z) ∈ R. In simpler terms, if x is related to y and y is related to z, then x must be related to z.
- Assume x R y and y R z for some x, y, z ∈ S.
- x R y means x and y were born in Delhi on the same day.
- y R z means y and z were born in Delhi on the same day.
- If x and y share the same birth day, and y and z share the same birth day, then x and z must necessarily share the same birth day.
- Therefore, if x R y and y R z hold, then x R z also holds for any x, y, z ∈ S.
The relation is transitive.
Conclusion: Type of Relation
A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
Since the given relation "were born in Delhi on the same day" is reflexive, symmetric, and transitive, it is an equivalence relation.
Let's examine the options:
- Option 1: The relation is an equivalence relation. This matches our conclusion.
- Option 2: The relation is not reflexive but it is symmetric and transitive. This is incorrect because we found it is reflexive.
- Option 3: The relation is not symmetric but it is reflexive and transitive. This is incorrect because we found it is symmetric.
- Option 4: The relation is not transitive but it is reflexive and symmetric. This is incorrect because we found it is transitive.
Therefore, the correct statement is that the relation is an equivalence relation.
Revision Table: Properties of Relations
| Property | Definition | Condition for R on Set S |
|---|---|---|
| Reflexive | Every element is related to itself. | For all x ∈ S, (x, x) ∈ R. |
| Symmetric | If x is related to y, then y is related to x. | For all x, y ∈ S, if (x, y) ∈ R, then (y, x) ∈ R. |
| Transitive | If x is related to y and y is related to z, then x is related to z. | For all x, y, z ∈ S, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R. |
| Equivalence Relation | A relation that is reflexive, symmetric, and transitive. | Satisfies all three properties listed above. |
Additional Information: Equivalence Classes
When a relation is an equivalence relation on a set S, it partitions the set S into disjoint subsets called equivalence classes. Each equivalence class consists of all elements that are related to each other.
In this specific example, the set S is the set of all persons living in Delhi, and the relation is being born in Delhi on the same day. The equivalence classes formed by this relation would group together all persons living in Delhi who share the exact same birthday (day, month, and year).
For instance, all persons in S born on January 1st, 1990, would form one equivalence class. All persons in S born on January 2nd, 1990, would form another equivalence class, and so on. These classes are disjoint because a person cannot be born on two different days.
Equivalence relations are fundamental in mathematics and are used to classify elements based on some shared property, effectively grouping them into categories.
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Consider the following relations on the set {1, 2, 3, 4}:
R1 = {(1, 1),(1, 2), (1, 4),(2, 1), (2, 2), (3, 3),(4, 1), (4, 4)}
R2 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
R3 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3 ), (2, 4), (3, 3), (3, 4), (4, 4)}
R4 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
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