A2 - Let C be the claim ± for all real numbers x 2 R...

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, f is a function from R to R . 1. For A a subset of R , (a) range , f ( A ) , of f over A . f ( A ) = f f ( x ) : x 2 A g (b) preimage , f 1 ( A ) , of A . f 1 ( A ) = f x : f ( x ) 2 A g 2. Show that if A and B are subsets of R , then f 1 ( A [ B ) = f 1 ( A ) [ f 1 ( B ) . To show that f 1 ( A [ B ) f 1 ( A ) [ f 1 ( B ) , suppose x 2 f 1 ( A [ B ) . This means that f ( x ) 2 A [ B , so f ( x ) 2 A or f ( x ) 2 B . If f ( x ) 2 A , then x 2 f 1 ( A ) , while if f ( x ) 2 B , then x 2 f 1 ( B ) . In either case, x 2 f 1 ( A ) [ f 1 ( B ) . To show that f 1 ( A [ B ) ± f 1 ( A ) [ f 1 ( B ) , suppose x 2 f 1 ( A ) [ f 1 ( B ) . Then x 2 f 1 ( A ) or x 2 f 1 ( B ) , which means that f ( x ) 2 A or f ( x ) 2 B . In either case, f ( x ) 2 A [ B , which means that x 2 f 1 ( A [ B ) . 3. Let f f ( x ) = x 2 . Let A be the closed interval [1 ; 2] . Describe f 1 ( A ) in terms of intervals . f 1 ( A ) = h ² p 2 ; ² 1 i [ h 1 ; p 2 i [1 ; 2] . 4.
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Unformatted text preview: Let C be the claim ± for all real numbers x 2 R , either x < or 1 2 x < x .² (a) Form the logical negation , D , of the claim C . Try to embed the word ± not ² as deeply into D as possible ( or avoid using it altogether ). There is a real number x such that x ³ and 1 2 x ³ x . (b) Which of the claims , C or D , is true , and why ? Claim D is true because, taking x = 0 , we see that ³ and 1 2 ³ ....
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This note was uploaded on 05/26/2010 for the course MATH maa 4200 taught by Professor Dr. during the Spring '08 term at Miami University.

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