s4 - Math 5615H: Introduction to Analysis I. Fall 2011...

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Unformatted text preview: Math 5615H: Introduction to Analysis I. Fall 2011 Homework #4. Problems and Solutions. #1. Let f be a mapping of A to B . Show that for each B 1 B and B 2 B , their inverse images satisfy the properties ( i ) f- 1 ( B 1 B 2 ) = f- 1 ( B 1 ) f- 1 ( B 2 ) , ( ii ) f- 1 ( B 1 B 2 ) = f- 1 ( B 1 ) f- 1 ( B 2 ) . Proof. (i) We have x f- 1 ( B 1 B 2 ) f ( x ) B 1 B 2 ( f ( x ) B 1 ) or ( f ( x ) B 2 ) ( x f- 1 ( B 1 ) ) or ( x f- 1 ( B 2 ) ) x f- 1 ( B 1 ) f- 1 ( B 2 ) . Therefore, both sides in (i) coincide. The proof of (ii) is quite similar, with being replaced by , and or by and. #2. Let f be a mapping of A to B . Verify whether of not the images of subsets A 1 A and A 2 A in general satisfy the properties ( i ) f ( A 1 A 2 ) = f ( A 1 ) f ( A 2 ) , ( ii ) f ( A 1 A 2 ) = f ( A 1 ) f ( A 2 ) ....
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s4 - Math 5615H: Introduction to Analysis I. Fall 2011...

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