Proving f(f-1(B))=B: Understanding Preimages and Set Inclusion

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In summary, the conversation discusses defining a function f from set S to set T, with B being a subset of T. The preimage of B, f-1(B), is also defined. The main question is whether to prove set equality using inclusion in both directions, and the suggested approach is to show that an element in the preimage of B maps to B. However, it is pointed out that the equality is not true in general and extra assumptions on the function f are needed. One possible assumption is that f is surjective.
  • #1
SMA_01
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Homework Statement



Define f:S→T, where B[itex]\subseteq[/itex]T. Let f-1(B)={x[itex]\in[/itex]S:f(x)[itex]\in[/itex]B} be the preimage of B.

Demonstrate that for any such map f, f(f-1(B))=B.

My main question is, would I prove this using set inclusion both ways?

I was going to begin by letting an element be in the preimage of B, and explain what that means, then mapping that element to B. Would this be correct?

I just need a push in the right direction.

Thank you.
 
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  • #2
SMA_01 said:

Homework Statement



Define f:S→T, where B[itex]\subseteq[/itex]T. Let f-1(B)={x[itex]\in[/itex]S:f(x)[itex]\in[/itex]B} be the preimage of B.

Demonstrate that for any such map f, f(f-1(B))=B.

My main question is, would I prove this using set inclusion both ways?

I was going to begin by letting an element be in the preimage of B, and explain what that means, then mapping that element to B. Would this be correct?

I just need a push in the right direction.

Thank you.
Generally, yes. To show set equality you need to show inclusion in both directions.
 
  • #3
Define f:S→T, where B[itex]\subseteq[/itex]T. Let f-1(B)={x[itex]\in[/itex]S:f(x)[itex]\in[/itex]B} be the preimage of B.

Demonstrate that for any such map f, f(f-1(B))=B.
Just to let you know, but that equality is false in general. You need extra assumptions on f.
 
  • #4
SammyS- Thank you

Micromass- Yes, the function needs to be surjective, right? I'm not sure why there were no assumptions...
 
  • #5
micromass said:
Just to let you know, but that equality is false in general. You need extra assumptions on f.
I'm very reluctant to question you, micromass.

But isn't the extra assumption of f needed for [itex]f^{-1}\left(f(A)\right)=A[/itex] rather than for [itex]f\left(f^{-1}(B)\right)=B\ ?[/itex]
 
  • #6
SammyS said:
I'm very reluctant to question you, micromass.

But isn't the extra assumption of f needed for [itex]f^{-1}\left(f(A)\right)=A[/itex] rather than for [itex]f\left(f^{-1}(B)\right)=B\ ?[/itex]

An extra assumption is needed for both. Consider [itex]f(x)=x^2[/itex] and B=[-1,0]. Then [itex]f^{-1}(B)=\{0\}[/itex] and thus [itex]f(f^{-1}(B))=\{0\}[/itex]. The problem is that f is not surjective.
 
  • #7
micromass said:
An extra assumption is needed for both. Consider [itex]f(x)=x^2[/itex] and B=[-1,0]. Then [itex]f^{-1}(B)=\{0\}[/itex] and thus [itex]f(f^{-1}(B))=\{0\}[/itex]. The problem is that f is not surjective.
Well, my reluctance was well founded!

Thanks as always.
 

FAQ: Proving f(f-1(B))=B: Understanding Preimages and Set Inclusion

What is the purpose of proving f(f-1(B))=B in a scientific study?

The purpose of proving f(f-1(B))=B is to establish the accuracy and validity of a mathematical function. This proof is essential in scientific research as it ensures that the results obtained from the function are reliable and can be used to make further conclusions or predictions.

How is f(f-1(B))=B proven in scientific studies?

Proving f(f-1(B))=B involves using mathematical techniques and logical reasoning to show that the composition of the function and its inverse results in the original set B. This can be done through algebraic manipulation, induction, or other methods depending on the specific function and its properties.

Why is it important to prove f(f-1(B))=B instead of assuming it to be true?

Proving f(f-1(B))=B is important because it eliminates any doubts or uncertainties about the accuracy of the function. By providing a rigorous proof, scientists can confidently use the function and its results in their research, without the risk of making incorrect assumptions.

Can f(f-1(B))=B be proven for all mathematical functions?

Yes, f(f-1(B))=B can be proven for all mathematical functions as long as the function is well-defined and its inverse exists. However, the methods and techniques used to prove it may vary depending on the complexity of the function.

Are there any real-life applications of proving f(f-1(B))=B?

Yes, proving f(f-1(B))=B has many real-life applications in various fields such as physics, engineering, and economics. It is particularly useful in validating and improving mathematical models used to describe real-world phenomena, as well as in designing experiments and analyzing data.

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