HW-2-solutions.pdf - York University—SC/MATH 1019 B Assignment 2 Solutions 1(2.3.42 modified(b Let f A → B and let X Y be subsets of the domain A

HW-2-solutions.pdf - York University—SC/MATH 1019 B...

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York University—SC/MATH 1019 B Assignment 2 Solutions October 10, 2019 1 (2.3.42, modified (b)) Let f : A B and let X, Y be subsets of the domain A . For any Z A , define the image of Z under f to be the set f [ Z ] = { b B |∃ z Z ( f ( z ) = b ) } . a. Show that f [ X Y ] = f [ X ] f [ Y ]. b. Give an example of a function f and subsets X, Y of its domain to show that it is not always true that f [ X Y ] = f [ X ] f [ Y ]. Solution (5 points each): (a): We have b f [ X Y ] if and only if there is x X Y so that f ( x ) = b if and only if there is x X so that f ( x ) = b or there is x Y so that f ( x ) = b if and only if b f [ X ] f [ Y ]. (b): (one possible example) Let A = { a 1 , a 2 } and B = { b } . Now define f : A B so that f ( a 1 ) = f ( a 2 ) = b . Then f [ { a 1 } ∩ { a 2 } ] = f [ ] = , but f [ { a 1 } ] f [ { a 2 } ] = { b } ∩ { b } = { b } . 2 (2.3.46) Let f : A B and let X, Y be subsets of the co-domain B . For any Z B , define the pre-image of Z under f to be the set f - 1 [ Z ] = { a A |∃ z Z ( f ( a ) = z ) } .
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