Proving an equivalence relation using inverse functions

In summary, for a function f : A → B and an equivalence relation Γ ⊂ B × B, the set (f × f)^-1 (Γ) ⊂ A × A is an equivalence relation on A. This can be described as {(a, a′) ∈ A × A|(f(a), f(a′)) ∈ Γ}. To prove this, we can show that for any (a, a′) ∈ (f × f)^-1 (Γ), it follows that (a, a′) also satisfies the definition of an equivalence relation. By showing that (f(a), f(a′)) ∈ Γ implies that (a, a′) satisfies the properties of an equivalence relation, we
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Homework Statement


Let f : A → B be a function and let Γ ⊂ B × B be an equivalence relation on B. Prove that the set (f × f)^-1 (Γ) ⊂ A × A (this can be described as {(a, a′) ∈ A × A|(f(a), f(a′)) ∈ Γ}) is an equivalence relation on A.

Homework Equations


The Attempt at a Solution


Let (f(a),f(a’)) ⊂ Γ. Since f(a) and f(a’) hold an equivalence relation with each other, it follows that a and a’ hold an equivalence relation with each other. Since f(a) and f(a’) are arbitrary elements of Γ, it follows that (fxf)-1Γ ⊂ A x A is an equivalence relation on A.

I'm not sure if thi is the right approach. In particular I am not sure that i can say that f(a) and f(a') holding an equivalence relation means that a and a' hold one too.
 
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Does anybody know if I am correct? I'm not sure if I'm skipping steps.
 

FAQ: Proving an equivalence relation using inverse functions

What is the definition of an equivalence relation?

An equivalence relation is a mathematical relationship between two elements, denoted by a~b, where a and b belong to a set, that satisfies three properties: reflexivity, symmetry, and transitivity.

How can inverse functions be used to prove an equivalence relation?

Inverse functions can be used to prove an equivalence relation by showing that the relationship between two elements is symmetric, meaning that if a~b then b~a. This can be done by finding the inverse function of the original function and showing that it satisfies the property of symmetry.

What is the role of reflexivity in proving an equivalence relation?

Reflexivity is one of the three properties that an equivalence relation must satisfy. It means that every element in a set must be related to itself, or in other words, a~a must be true for all elements a in the set. Inverse functions can help prove reflexivity by showing that the function is its own inverse.

Can inverse functions be used to prove transitivity in an equivalence relation?

Yes, inverse functions can also be used to prove transitivity, which is another key property of an equivalence relation. Transitivity means that if a~b and b~c, then a~c. By using the inverse function of the original function, it can be shown that if the function maps a to b and b to c, then it must also map a to c, satisfying the property of transitivity.

Is it necessary for an equivalence relation to have an inverse function?

No, it is not necessary for an equivalence relation to have an inverse function. Inverse functions are simply a helpful tool in proving the properties of an equivalence relation, but they are not a requirement. There are other methods for proving an equivalence relation, such as using composition of functions or using logical arguments.

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