Inverse map is closed under complementation

In summary, we must prove the equality $f^{-1}(E^c) = (f^{-1}(E))^c$, where $f$ is a map from $X$ to $Y$ and $E$ is a subset of $Y$. To do this, we will show that each set is contained in the other. This involves proving that $f^{-1}(E^c) \subseteq (f^{-1}(E))^c$ and $(f^{-1}(E))^c \subseteq f^{-1}(E^c)$. We will do this by taking an element from one set and showing that it is also in the other set. For the first inclusion, let $x \in f^{-1}(
  • #1
carr1
2
0
f^-1 (E^c) = (f^-1(E))^c where f is map from X to Y and E is in Y.
Prove equality is true.
 
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  • #2
how would I show the inverse map on the left is a subset of the inverse map on the right? and vice versa?
 
  • #3
To show two sets are equal we show each is contained in the other, hence we must show $f^{-1}(E^c) \subseteq (f^{-1}(E))^c$ and $(f^{-1}(E))^c \subseteq f^{-1}(E^c)$. To do this we take an element in one of them and show it is also in the other. I'm going to do the first inclusion.

Let $x \in f^{-1}(E^c)$. By the definition of inverse image we know that $f(x) \in E^c$, but this means that $f(x) \notin E$. Hence $x \notin f^{-1}(E)$ and we conclude that $x \in (f^{-1}(E))^c$. Therefore $f^{-1}(E^c) \subseteq (f^{-1}(E))^c$.

Try the second inclusion. :)

Best wishes,

Fantini.
 

Related to Inverse map is closed under complementation

What is an inverse map?

An inverse map is a function that reverses the inputs and outputs of another function. In other words, if a function maps an input to an output, the inverse map will map that output back to the original input.

What is closure under complementation?

Closure under complementation is a property of sets where if a set is closed under some operation, its complement (the elements not in the set) is also closed under the same operation.

What does it mean for inverse map to be closed under complementation?

If an inverse map is closed under complementation, it means that if a set of inputs is mapped to a set of outputs, the inverse map will also map the complement of those outputs back to the complement of the inputs.

Why is it important for the inverse map to be closed under complementation?

This property is important because it ensures that the inverse map is a valid function, meaning it has a unique output for every input. It also allows for more flexibility and ease of use in mathematical and scientific applications.

Are there any other important properties of inverse maps?

Yes, another important property is closure under composition, which means that if you compose the inverse map with another function, the resulting function is also an inverse map. This allows for the use of inverse maps in more complex mathematical operations.

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