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Homework 7

# Homework 7 - MATH 74 HOMEWORK 7(DUE WEDNESDAY OCTOBER 24...

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MATH 74 HOMEWORK 7 (DUE WEDNESDAY OCTOBER 24) 1(a). There are four functions with domain A = { 1 , 2 } and codomain B = { 1 , 2 } . Write them down. (I would do this by giving each a name and defining them explicitly; for example “there is f : A B given by f (1) = · · · and f (2) = · · · , and g : A B given by...”) 1(b). How many functions f : A B have the property that f ( f ( x )) = f ( x ) for all x A ? 2. Let C = { 1 , 2 , 3 } . Whatever your answer was for 1(b), write down more than that number of functions f : C C with the property that f ( f ( x )) = f ( x ) for all x C . 3. If f : R R is a function satisfying f ( f ( x )) = x, x R , is it necessarily true that f ( x ) = x for all x R ? Prove this, if it is true, and give a counterexample if it is false. 4(a). Let N = { 1 , 2 , . . . } and let D = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . Define a function g : N D by g ( n ) = the “ones digit” of 3 n Thus for example g (4) = 1 because 3 4 = 81. Explicitly identify the range of g . What I mean by this, is list the elements of a subset R of D with the property that R is precisely the set { g ( n ) : n N } .

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