Question

What is equivalence class class 12?

Answer 1

An equivalence class is a concept from mathematics, specifically in the branch of set theory and relations. It's a way of grouping elements of a set that are considered "equivalent" to each other under a particular relation.

To understand this, let’s break it down step-by-step:

1. Equivalence Relation

An equivalence relation is a special type of relation on a set. It must satisfy three key properties:

1. Reflexive: Every element is related to itself. In other words, for any element \( a \) in set \( A \), \( a \sim a \).
   

  

1. Symmetric: If one element is related to another, the second is also related to the first. For example, if \( a \sim b \), then \( b \sim a \).
   

  

1. Transitive: If one element is related to a second, and the second is related to a third, then the first element is related to the third. For example, if \( a \sim b \) and \( b \sim c \), then \( a \sim c \).
   

2. Equivalence Class

Once we have an equivalence relation, we can group all the elements that are equivalent to each other. This group is called an equivalence class.

For a given element \( a \) in a set \( A \), the equivalence class of \( a \), denoted by \( [a] \), consists of all elements in \( A \) that are related to \( a \) under the equivalence relation.

In simple words:

1. An equivalence class is the set of all elements that are considered "equal" to a given element based on the equivalence relation.
   

Example:

Consider the set of integers \( \mathbb{Z} \), and let’s define a relation \( \sim \) where two numbers are related if their difference is divisible by 2. That is:

1. \( a \sim b \) if and only if \( a - b \) is divisible by 2.
   

This relation is an equivalence relation because it is:

1. Reflexive: \( a - a = 0 \), which is divisible by 2.
   

1. Symmetric: If \( a - b \) is divisible by 2, then \( b - a \) is also divisible by 2.
   

1. Transitive: If \( a - b \) and \( b - c \) are divisible by 2, then \( a - c \) is also divisible by 2.
   

Now, the equivalence classes would be:

1. One class for even numbers: \( [0] = \{0, 2, -2, 4, -4, \dots \} \)
   

1. One class for odd numbers: \( [1] = \{1, -1, 3, -3, 5, -5, \dots \} \)
   

So, the equivalence classes divide the set \( \mathbb{Z} \) into two groups: the set of even integers and the set of odd integers.

Summary:

1. Equivalence class: A subset of elements in a set that are all equivalent to each other under a particular equivalence relation.
   

1. It's a way of grouping elements that share a common property, as defined by the equivalence relation.
   

Does that help clarify what an equivalence class is?