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Homework Statement
Given is the set X. The set of functions from X to [0,1] we call Fun(X,[0,1]). On this set we consider the relation R. An ordered pair (f,g) belongs to R when [tex]f^{-1}(0)\setminus g^{-1}(0)[/tex] is a countable set.
a) Prove that R is transitive.
b) Is R an equivalence relation? Prove!
c) Prove that [tex]R \cap R^{-1}[/tex] is an equivalence relation.
Homework Equations
Transitive means that if (f,g) and (g,h) belong to the relation, that also (f,h) belongs to it.
Equivalence relation is a relation that is transitive, reflexive ([tex](f,f) \in R[/tex] and symmetric ([tex](f,g) \in R \Rightarrow (g,f) \in R[/tex].
The Attempt at a Solution
a) [tex]f^{-1}(0)\setminus g^{-1}(0)[/tex] is a countable set. So [tex]f{-1}(0)[/tex] is a countable set. This means that [tex]f^{-1}(0)\setminus h^{-1}(0)[/tex], because a subset of a countable set is also countable.
Is this correct?
b) Reflexivity is easy, because [tex]f^{-1}(0)\setminus f^{-1}(0)[/tex] is the empty set so that is obviously countable.
How do I prove that it is symmetric?
c) Is this just the subset of all the reflexive pairs?
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