What is equivalence class class 12?
2 Answers
An equivalence class in mathematics, particularly in the context of class 12 studies, refers to a subset of a set formed by an equivalence relation. An equivalence relation is a relation that is reflexive, symmetric, and transitive. For example, in the set of integers with the relation of congruence modulo \( n \), two integers are equivalent if their difference is divisible by \( n \). The equivalence class of an element \( a \) under this relation is the set of all integers that are equivalent to \( a \), meaning all integers that, when divided by \( n \), have the same remainder as \( a \). This concept is fundamental in abstract algebra and number theory. Would you like more details on how to apply this concept?
In Class 12 mathematics, an equivalence class refers to a concept in set theory and mathematical logic that arises from the notion of equivalence relations. Here's a breakdown:
1. **Equivalence Relation:** This is a relationship between elements of a set that satisfies three properties:
- **Reflexivity:** Every element is related to itself.
- **Symmetry:** If one element is related to another, then the second element is related to the first.
- **Transitivity:** If one element is related to a second element, and that second element is related to a third element, then the first element is related to the third.
2. **Equivalence Class:** Given an equivalence relation on a set, an equivalence class is a subset of elements that are all related to each other by the equivalence relation. For any element in the set, the equivalence class of that element includes all elements that are related to it.
For example, if we have a set \( A \) and an equivalence relation \( \sim \) on \( A \), then the equivalence class of an element \( a \in A \) is defined as:
\[
[a] = \{ x \in A \mid x \sim a \}
\]
This means \( [a] \) is the set of all elements in \( A \) that are equivalent to \( a \).
3. **Example:** Consider the set of integers \( \mathbb{Z} \) and the equivalence relation "having the same parity" (i.e., being either both even or both odd). The equivalence classes under this relation are:
- The set of all even integers.
- The set of all odd integers.
In this case, the equivalence class of any even integer would be the set of all even integers, and the equivalence class of any odd integer would be the set of all odd integers.
Equivalence classes partition the set into disjoint subsets where each subset contains elements that are considered equivalent under the given relation.
1. **Equivalence Relation:** This is a relationship between elements of a set that satisfies three properties:
- **Reflexivity:** Every element is related to itself.
- **Symmetry:** If one element is related to another, then the second element is related to the first.
- **Transitivity:** If one element is related to a second element, and that second element is related to a third element, then the first element is related to the third.
2. **Equivalence Class:** Given an equivalence relation on a set, an equivalence class is a subset of elements that are all related to each other by the equivalence relation. For any element in the set, the equivalence class of that element includes all elements that are related to it.
For example, if we have a set \( A \) and an equivalence relation \( \sim \) on \( A \), then the equivalence class of an element \( a \in A \) is defined as:
\[
[a] = \{ x \in A \mid x \sim a \}
\]
This means \( [a] \) is the set of all elements in \( A \) that are equivalent to \( a \).
3. **Example:** Consider the set of integers \( \mathbb{Z} \) and the equivalence relation "having the same parity" (i.e., being either both even or both odd). The equivalence classes under this relation are:
- The set of all even integers.
- The set of all odd integers.
In this case, the equivalence class of any even integer would be the set of all even integers, and the equivalence class of any odd integer would be the set of all odd integers.
Equivalence classes partition the set into disjoint subsets where each subset contains elements that are considered equivalent under the given relation.