Macroeconomics Exam Review 65

Macroeconomics Exam Review 65 - 2.27 The identity function...

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Unformatted text preview: 2.27 The identity function : → is defined by ( ) = for every ∈ . Therefore 2 ≻ 1 = ⇒ ( 2 ) = 2 ≻ 1 = ( 1 ) 2.28 Assume that and are increasing. Then, for every 1 , 2 ∈ with 2 ≿ 1 ( 2 ) ≿ ( 1 ) = ⇒ ( ( 2 )) ≿ ( ( 1 )) ∘ is also increasing. Similarly, if and are strictly increasing, 2 ≻ 1 = ⇒ ( 2 ) ≻ ( 1 ) = ⇒ ( ( 2 )) ≻ ( ( 1 )) 2.29 For every ∈ ( ), there exists a unique ∈ such that ( ) = y. (For if 1 , 2 are such that ( 1 ) = ( 2 ), then 1 = 2 .) Therefore, is one-to-one and onto ( ), and so has an inverse (Exercise 2.4). Further 2 > 2 ⇐⇒ (...
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