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Unformatted text preview: MATH 347: SOLUTIONS FOR MIDTERM 1 NOTE: 100% = 40 points. 1 (10 points) : Suppose C and D are subsets of the domain of a function f . (a) Prove that f ( C ∩ D ) ⊂ f ( C ) ∩ f ( D ). By the definition of a subset, we have to show that any element of f ( C ∩ D ) is also an element of f ( C ) ∩ f ( D ). To this end, suppose x ∈ f ( C ∩ D ). Then there exists a ∈ C ∩ D s.t. f ( a ) = x . As a ∈ C , we have x = f ( a ) ∈ f ( C ). Similarly, x ∈ f ( D ). Thus, x ∈ f ( C ) ∩ f ( D ). (b) Give an example showing that the inlusion in part (a) may be strict. It may happen that f ( C ∩ D ) is a strict subset of f ( C ) ∩ f ( D ). To provide an example, consider U = { 1 , 2 , 3 } , C = { 1 , 3 } , and D = { 2 , 3 } . Define the function f : U → N by setting f (1) = f (2) = 5 and f (3) = 10. Then f ( C ) = f ( D ) = { 5 , 10 } , hence f ( C ) ∩ f ( D ) = { 5 , 10 } . On the other hand, C ∩ D = { 3 } , hence f ( C ∩ D ) = { f (3) } = { 10 } ....
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This note was uploaded on 01/27/2010 for the course MATH 347 taught by Professor ? during the Spring '09 term at University of Illinois at Urbana–Champaign.
 Spring '09
 ?
 Math, Sets

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