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Functions of (AUB)

  • Thread starter mtayab1994
  • Start date
  • #1
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Homework Statement


Let g be a function from ℝ to ℝ and for all subsets A and B of R.

Homework Equations


Prove that:
[tex]g^{-1}(A\cup B)=g^{-1}(A)\cap g^{-1}(B)[/tex]
and
[tex]g^{-1}(A\cap B)=g^{-1}(A)\cup g^{-1}(B)[/tex]


The Attempt at a Solution


Our teacher gave us this problem but I think it's wrong because I was easily able to prove that:
[tex]g^{-1}(A\cup B)=g^{-1}(A)\cup g^{-1}(B)[/tex]
and
[tex]g^{-1}(A\cap B)=g^{-1}(A)\cap g^{-1}(B)[/tex]
but when i try to solve what he gave us this is all I can do:

[tex]x\in g^{-1}(A\cup B)\Longleftrightarrow g(x)\in A\cup B\Longleftrightarrow g(x)\in Aorg(x)\in B\Longleftrightarrow x\in g^{-1}(A)\cup g^{-1}(B)[/tex]

and the same goes for the second one. Can anyone tell me where I'm going wrong?
 

Answers and Replies

  • #2
pasmith
Homework Helper
1,740
412

Homework Statement


Let g be a function from ℝ to ℝ and for all subsets A and B of R.

Homework Equations


Prove that:
[tex]g^{-1}(A\cup B)=g^{-1}(A)\cap g^{-1}(B)[/tex]
and
[tex]g^{-1}(A\cap B)=g^{-1}(A)\cup g^{-1}(B)[/tex]
These are wrong. A counterexample to both is the zero function [itex]g(x) \equiv 0[/itex] with [itex]A = \{0\}[/itex] and [itex]B = \{1\}[/itex].

The Attempt at a Solution


Our teacher gave us this problem but I think it's wrong because I was easily able to prove that:
[tex]g^{-1}(A\cup B)=g^{-1}(A)\cup g^{-1}(B)[/tex]
and
[tex]g^{-1}(A\cap B)=g^{-1}(A)\cap g^{-1}(B)[/tex]
These are correct.
 

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