Macroeconomics Exam Review 57

# Macroeconomics Exam Review 57 - Chapter 2 Functions 2.1 In...

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Unformatted text preview: Chapter 2: Functions 2.1 In general, the birthday mapping is not one-to-one since two individuals may have the same birthday. It is not onto since some days may be no one’s birthday. 2.2 The origin is fixed point for every . Furthermore, when = 0, is an identity function and every point is a fixed point. 2.3 For every ∈ , there exists some ∈ such that ( ) = , whence ∈ − 1 ( ). Therefore, every belongs to some contour. To show that distinct contours are disjoint, assume ∈ − 1 ( 1 ) ∩ − 1 ( 2 ). Then ( ) = 1 and also ( ) = 2 . Since is a function, this implies that 1 = 2 . 2.4 Assume is one-to-one and onto. Then, for every ∈ , there exists ∈ such that ( ) = . That is, − 1 ( ) ∕ = ∅ for every ∈ . If is one to one, ( ) = = ( ′ ) implies...
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