Homework 7 Solution

Homework 7 Solution - MATH 74 HOMEWORK 7 (DUE WEDNESDAY...

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MATH 74 HOMEWORK 7 (DUE WEDNESDAY OCTOBER 17) 1(a). There is f : A B given by f (1) = 1 and f (2) = 1, and g : A B given by g (1) = 1 and g (2) = 2, and h : A B given by h (1) = 2 and h (2) = 1, and k : A B given by k (1) = 2 and k (2) = 2. 1(b). Three of them do. (As I named them, f , g , and k ; h doesn’t.) 2. Consider e.g. f 1 : C C given by f 1 ( x ) = 1 for all x , and f 2 : C C given by f 2 ( x ) = 2 for all x , and f 3 : C C given by f 3 ( x ) = 3 for all x , and f 4 : C C given by f 4 ( x ) = x for all x . [Other solutions possible.] 3. It is not necessarily true that f ( x ) = x for all x R . Consider for example the function f : R R given by x 7→ - x . Then f ( f ( x )) = - ( - x )) = x for all x R , but f (2) = - 2 is not 2. [Other solutions possible.] 4(a). The range of g is the set R = { 1 , 3 , 7 , 9 } . 4(b). We first prove R Im g . To see this, note that 1 Im g because g (4) = 1 (as 3 4 = 81), that 3 Im g because g
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Homework 7 Solution - MATH 74 HOMEWORK 7 (DUE WEDNESDAY...

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