This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 10. OCTOBER 5TH: FUNCTIONS. 25 10. October 5th: Functions. 10.1 . Let A , B be sets, and f : A B a function. Let S and T be subsets of A . Is f ( S T ) necessarily equal to f ( S ) f ( T )? Give a proof or find a coun terexample. Is f ( S T ) necessarily equal to f ( S ) f ( T )? Proof or counterexample. Answer: f ( S T ) and f ( S ) f ( T ) need not be equal. For example, consider the function f : { , *} { } defined by f ( ) = 0, f ( * ) = 0, and let S = {} , T = {*} . Then S T = , so f ( S T ) = ; while f ( S ) f ( T ) = { } = . Note that the inclusion f ( S T ) f ( S ) f ( T ) always holds. (Why?) It is true that f ( S ) f ( T ) = f ( S T ) for all A, B, f, S, T . Proof: Since S and T are subsets of S T , then f ( S ) and f ( T ) are both subsets of f ( S T ); this shows f ( S ) f ( T ) f ( S T ). On the other hand, if b f ( S T ) then a S T such that f ( a ) = b . Since a S T , then a S or a T . If a S , then...
View Full
Document
 Fall '11
 Aluffi

Click to edit the document details