Is all function a equvalence relation?

Thread starter np infinite
Start date Feb 23, 2013
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Function Relation

In summary, an equivalence relation is a mathematical concept that describes a relationship between elements of a set. It satisfies three properties: reflexivity, symmetry, and transitivity. A function, on the other hand, is a mathematical rule that associates one input value with exactly one output value. It can be seen as a special type of equivalence relation where the input value is equivalent to the output value. However, not all functions can be considered as equivalence relations as they must satisfy the three properties. It is important to understand the relationship between function and equivalence relation in order to better understand the properties and behavior of functions and to apply them to different areas of mathematics.

Feb 23, 2013

np infinite

I know that function is a type of relation. What i acctually need to know is that; Are these relations Equivalent all the times?, or it depends on the types of function?

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Feb 23, 2013

micromass

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Well, what do you think yourself?? Take an arbitrary function such as [itex]f:\mathbb{R}\rightarrow \mathbb{R}x\rightarrow x^2[/itex]. Does this induce an equivalence relation??

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that describes a relationship between elements of a set. It is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity.

2. What is a function?

A function is a mathematical rule that associates one input value with exactly one output value. It can be thought of as a machine that takes in an input and produces a corresponding output.

3. How is a function related to an equivalence relation?

A function can be seen as a special type of equivalence relation where the input value is equivalent to the output value. This means that a function is reflexive, symmetric, and transitive.

4. Can all functions be considered as equivalence relations?

No, not all functions can be considered as equivalence relations. In order for a function to be an equivalence relation, it must satisfy the three properties: reflexivity, symmetry, and transitivity. If a function does not satisfy these properties, it cannot be considered as an equivalence relation.

5. Why is it important to understand the relationship between function and equivalence relation?

Understanding the relationship between function and equivalence relation is important in mathematics because it helps us to better understand the properties and behavior of functions. It also allows us to generalize certain concepts and apply them to different areas of mathematics.