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Unformatted text preview: Math 350 Advanced Calculus I Fall 2009 Homework Solutions Section 8 CARDINALITY Text Problems 8.3 Show that each of the following pairs of sets S and T are equinumerous by finding a specific bijection between them. (a) S = [0 , 1] and T = [1 , 3] We need a function that maps the interval S = [0 , 1] onto the interval T = [1 , 3]. Define f : S T by f ( x ) = 2 x + 1 for each x S = [0 , 1]. We need to show that f is both injective and surjective. Injective. To show that f is injective, we need to show ( x 1 S ) ( x 2 S ) [ f ( x 1 ) = f ( x 2 ) x 1 = x 2 ] . Suppose x 1 and x 2 are arbitrary elements in S = [0 , 1]. Suppose f ( x 1 ) = f ( x 2 ) . We need to show x 1 = x 2 . Since f ( x 1 ) = f ( x 2 ), then 2 x 1 + 1 = 2 x 2 + 1 which implies x 1 = x 2 , as required. Therefore f is injective. Surjective. To show that f is surjective, we need to show ( y T ) ( x S ) [ f ( x ) = y ] ....
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