s5 - Math 55 Worksheet Adapted from worksheets by Rob...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Sets, Functions, Cardinality 1. Determine whether each of the following are injective, surjective, both (bijective), or neither. Prove your answer (a) f : N N f ( x ) = x 2 Injective (b) f : R R f ( x ) = 3 x + 4 Bijective (c) f : (0 , 1) (1 , ) f ( x ) = 1 x Biective 2. For a function f : A B , given S A , let f ( S ) := { y B : x S,f ( x ) = y } . Prove that if f is one-to-one, then f ( S T ) = f ( S ) f ( T ) for any S,T A . We have to show that the two SETS f ( S T ) and f ( S ) f ( T ) are equal. So we have two steps: 1) We show f ( S T ) f ( S ) f ( T ): Let b f ( S T ). Then this means that there exists an x S T such that b = f ( x ). Thus x S and x T . So f ( x ) f ( S ) and f ( x ) f ( T ) Therefore f ( x ) f ( T ) f ( S ). But
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s5 - Math 55 Worksheet Adapted from worksheets by Rob...

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