Share this:

If x is a subset of Y then, is inverse of x also a subset of Y?

Let $ f : X \rightarrow Y $ be a function. If B is subset of y then its inverse image $ f^-B $ is the subset of x define by $ f^-B = {x : f(x) \in B} $

Now prove the following.

$ i. B_1 \subset B_2 \Rightarrow f^-B_1 \subset f^- B_1 $

$ ii. f^-\cup_i B_i = \cup_i f^-B_i $

$ iii. f^-\cap_i B_i \subset \cap_i f^-B_i $

Solution:

Let $ F(x) : x \rightarrow y $ be a function. If B is the subset of Y, then it's inverse maps

$ f^-x $ is the subset of x define by $ f^-B = { f(x) \in B} $

i.

$ let\ x \in f^-B_1\ then\ f(x) \in B_1 $

$ since\ B_1 \subset B_1\ $ $ we\ can\ write,\ f(x) \in f^-(B_2) $

ii.

$ Suppose\ , x \in f^-\cup_i(B_i) $, $ f(X) \in \cup_i(B_i) $

$ So\ f(x) \in B_i for\ some\ i $ this implies that, $ x \in f^-B_i $

$ So\ x \in \cap_i f^-(B_i)$.

Conversely

$ Suppose\ x \in \cap_i f^-(B_i) $. $ Then\ x \in f^-(B_i)\ for\ some\ i $

$ So\ f(x) \in B_i \subset (\cap_iB_i) $ therefore $ f(x) \in (\cap_i B_i) $ and hence $ x \in f^-(\cap_iB_i)$

Leave a Reply

Share this:

Subscribe to our Newsletter

Hello surfer, thank you for being here. We are as excited as you are to share what we know about data. Please subscribe to our newsletter for weekly data blogs and many more. If you’ve already done it, please close this popup.



No, thank you. I do not want.
100% secure.
Scroll to Top