Answered: How do you show the inverse image of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

How do you show the inverse image of the intersecction of S and T equals the intersection of the image of S and the image of T if f is a function from A to B and S And T are subsets of A?

Expert Solution

Step 1

To prove : $f^{- 1} (S \cap T) = f^{- 1} (S) \cap f^{- 1} (T)$

To show $f^{- 1} (S \cap T) = f^{- 1} (S) \cap f^{- 1} (T)$ need to prove $f^{- 1} (S \cap T) \subseteq f^{- 1} (S) \cap f^{- 1} (T)$ and $f^{- 1} (S) \cap f^{- 1} (T) \subseteq f^{- 1} (S \cap T)$ .

First prove $f^{- 1} (S \cap T) \subseteq f^{- 1} (S) \cap f^{- 1} (T)$ .

To prove $f^{- 1} (S \cap T) \subseteq f^{- 1} (S) \cap f^{- 1} (T)$ show the below condition :

If $a \in f^{- 1} (S \cap T)$ then $a \in f^{- 1} (S) \cap f^{- 1} (T)$

Let $x \in f^{- 1} (S \cap T)$ then target is to show $x \in f^{- 1} (S) \cap f^{- 1} (T)$ .

$\begin{array}{rcl} x & \in & f^{- 1} (S \cap T) \\ f (x) & \in & S \cap T \\ f (x) & \in & S and f (x) \in T \\ x & \in & f^{- 1} (S) and x \in f^{- 1} (T) \\ x & \in & f^{- 1} (S) \cap f^{- 1} (T) \end{array}$

Hence $f^{- 1} (S \cap T) \subseteq f^{- 1} (S) \cap f^{- 1} (T)$ .

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