Suppose that A is a nonempty set, and f is a function that...

Suppose that $A$ is a nonempty set, and $f$ is a function that has $A$ as its domain. Let $R$ be the relation on $A$ consisting of all ordered pairs $(x, y)$ such that $f(x)=f(y)$ .
a) Show that $R$ is an equivalence relation on $A$ .
b) What are the equivalence classes of $R ?$

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Key Concepts

Function-induced Equivalence Relation

A function-induced equivalence relation is defined by relating two elements if they have the same image under a given function. This concept is fundamental in abstract algebra and set theory as it provides a method to partition a set based on the values taken by a function, leading to a natural equivalence partition that reflects the structure of the function's mapping.

Equivalence Class

An equivalence class under an equivalence relation is a subset of the set containing all elements that are equivalent to each other. When a function induces an equivalence relation by having equal outputs, each equivalence class consists of all elements in the domain that map to a particular value in the codomain, effectively partitioning the set into distinct groups.

Equivalence Relation

An equivalence relation on a set is a relationship that is reflexive, symmetric, and transitive. In this context, demonstrating that the relation defined by 'f(x) = f(y)' has these properties shows that it satisfies the criteria for an equivalence relation, meaning every element relates to itself, the relation is mutual between any two related elements, and the relation can be extended transitively across elements.

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