Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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What is the correct answer to the blank that I boxed?

Let X and Y be any sets, and let F be any onto function from X to Y.
Prove that for every subset B C Y, F(F¯1 (B))
= B.
Proof: To prove that F(F (B))
B, we must show that F(F`
(B)) C B and that B C F(F¯1 (B)).
Proof that F(F¯1 (B)) C B: Suppose y E F(F1(B)). We must show that y e B
[we
By definition of image of a set, there exists an element x in F-(B)
such that F(x) = y
. In addition, because x is in F-(B)
, by definition of
inverse image, F(x) E B
Therefore, by substitution, y € B
Proof that B C F(F¯1 (B)): Suppose y E B. We must show that y is in F(F-"(B))
It follows by definition
Because F is onto, there exists an x in X, such that F(x) = y
of image of a set that F(x) E F-(B)
So, since y E B and by definition of an inverse image, x E F-(B)
Therefore, by substitution, y E F(F-(B))
Conclusion: Since both subset relations have been proved, we conclude that F(F (B))
= B.
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Transcribed Image Text:Let X and Y be any sets, and let F be any onto function from X to Y. Prove that for every subset B C Y, F(F¯1 (B)) = B. Proof: To prove that F(F (B)) B, we must show that F(F` (B)) C B and that B C F(F¯1 (B)). Proof that F(F¯1 (B)) C B: Suppose y E F(F1(B)). We must show that y e B [we By definition of image of a set, there exists an element x in F-(B) such that F(x) = y . In addition, because x is in F-(B) , by definition of inverse image, F(x) E B Therefore, by substitution, y € B Proof that B C F(F¯1 (B)): Suppose y E B. We must show that y is in F(F-"(B)) It follows by definition Because F is onto, there exists an x in X, such that F(x) = y of image of a set that F(x) E F-(B) So, since y E B and by definition of an inverse image, x E F-(B) Therefore, by substitution, y E F(F-(B)) Conclusion: Since both subset relations have been proved, we conclude that F(F (B)) = B.
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