109_sp2011_ho_injsurj_sol

109_sp2011_ho_injsurj_sol - 109 Spring 2011 Injections and...

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109 Spring 2011 - Injections and Surjections Exercise (II.12) . Suppose that A Z . Write the following statement entirely in symbols using the quantifiers and . Write out the negative of this statement in symbols. There is a greatest number in the set A . Give an example of a set A for which this statement is true. Give an example of a set A for which it is false. a A x A ( x a ) If A = { 1 , 2 , 5 } then the statement is true. If A = Z then the statement is false. Exercise (II.17) . Functions f : R R and g : R R are defined as follows. f ( x ) = x + 2 if x < 1, x if 1 x 1 x 2 if x > 1; g ( x ) = x 2 if x < 1, x if 1 x 1 x + 2 if x > 1 Find the functions f g , g f . Is g the inverse of f ? Is f injective or surjective? Is g ? f g ( x ) = x ; g f ( x ) = x if x < 3, ( x + 2) if 3 x < 1 x if 1 x 1 ( x 2) if 1 < x 3 x if x > 3 Since g f is not the identity map, g
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