Equivalence Classes Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium

Define

f \sim g

on functions from

\mathbb{R}

\mathbb{R}

f \sim g

iff

f(0) = g(0)

. Verify this is an equivalence relation and describe the equivalence class of

f(x) = x^2

Solution

1
Reflexive: $f(0)=f(0)$ . True. Symmetric: if $f(0)=g(0)$ then $g(0)=f(0)$ . True. Transitive: if $f(0)=g(0)$ and $g(0)=h(0)$ , then $f(0)=h(0)$ . True.
2
The equivalence class of $f(x)=x^2$ (where $f(0)=0$ ): all functions $g$ with $g(0)=0$ , e.g., $g(x)=x, x^3, \sin x, 5x^2+3x$ .

Answer

[x^2] = \{g : \mathbb{R}\to\mathbb{R} \mid g(0)=0\}

Equivalence classes can be defined on any set. Here, functions are grouped by their value at

x=0

. The class

[x^2]

contains all functions that vanish at the origin.

About Equivalence Classes

An equivalence class is the set of all elements that are related to a given element under an equivalence relation — it groups objects that are considered 'the same' in some specified sense.

Learn more about Equivalence Classes →

More Equivalence Classes Examples

Example 1 medium

Define the relation [formula] on [formula] by '[formula] iff [formula]'. Verify this is an equivalen

Example 2 medium

On the set of triangles, define [formula] if [formula] is congruent to [formula]. Show this is an eq

Example 3 easy

List the equivalence classes of [formula] under [formula] iff [formula].

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Related Concepts

Set Equivalence Abstraction