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Midterm1SolutionsFA09 - MATH 347 SOLUTIONS FOR MIDTERM 1...

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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 } . 2 (10 points) : For x R , consider the following statements: (i) P : 0 < x 2 < 3; (ii) Q : | x | = 1; (iii) R : x 2 / Z ; (iv) S : x 2 = 2. Which of the following statements are true for any x R ?
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