MHB Inverse map is closed under complementation

carr1
Messages
2
Reaction score
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.
 
Physics news on Phys.org
how would I show the inverse map on the left is a subset of the inverse map on the right? and vice versa?
 
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.
 

Similar threads

Replies
5
Views
611
Replies
43
Views
5K
Replies
2
Views
2K
Replies
3
Views
2K
Replies
3
Views
1K
Replies
2
Views
2K
Back
Top