Proving a known proof involving injection.

In summary, in order to prove f^-1(f(E)) = E, you need to show that for any x in E, it is also in f^-1(f(E)), and vice versa. This can be done by using similar arguments in both directions and showing that the two sets are contained in each other, thus proving their equality.
  • #1
grjmmr
2
0
trying to learn how to do proofs. So I have A=> B which is injective and E [itex]\subseteq[/itex] B then prove f^-1(f(E)) = E.
So let x [itex]\in[/itex] f^-1(f(E)) => thus f(x) [itex]\in[/itex] f(E) => x[itex]\in[/itex] E

So I have proved that x is a point within E, a subset of A, to me I think I am missing something and have not proved f^-1(f(E)) = E.

any suggestions?
 
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  • #2
You need to do a similar argument in the opposite direction. Take x in E and show

it is in f-1f(E). You have then showed, for the two sets:

E is contained in f-1f(E).

and

f-1f(E) is contained in E.

This is the standard way of showing equality of sets.
 

1. What is the purpose of proving a known proof involving injection?

The purpose of proving a known proof involving injection is to provide a rigorous and logical demonstration of the validity of a mathematical statement or theory. This helps to strengthen the understanding and acceptance of the proof by others in the scientific community.

2. How do you determine if a proof involving injection is valid?

A proof involving injection is considered valid if it follows the established rules and principles of mathematical logic, and if all of its steps can be clearly and logically justified. This typically involves using axioms, definitions, and previously proven theorems to build a logical argument.

3. What is the difference between an injection and a surjection?

An injection is a function that maps each element of its domain to a unique element in its range, while a surjection is a function that maps its entire range onto its entire codomain. In other words, an injection is a one-to-one mapping, while a surjection is an onto mapping.

4. Can a proof involving injection be used to prove other mathematical statements?

Yes, a proof involving injection can often be used as a tool to prove other mathematical statements. This is because the concept of injection is closely related to other important concepts in mathematics, such as bijections and isomorphisms.

5. What are some real-world applications of proofs involving injection?

Proofs involving injection have numerous applications in fields such as computer science, physics, and economics. For example, they can be used to prove the correctness of algorithms, to show the isomorphism of physical systems, or to demonstrate the existence of optimal solutions in economic models.

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