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Unformatted text preview: Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation AD, Spring 2008 6 Functions and Bijections 6.1 Functions READ: Textbook pages 125132. EXERCISES: Pages 153161, #117, 8185. Recall from your previous mathematical knowledge that a function assigns exactly one output to each input. This is a special type of relation: Definition 6.1 Let A and B be sets. A function f from A to B (or, from A into B ), de- noted f : A B , is a relation from A to B that satisfies (i) Dom ( f ) = A (ii) If ( x,y ) f and ( x,z ) f , then y = z . In the case where A = B , we say f is a function on A . The function f is also called a mapping . We also refer to B as the codomain of f . Note that Rng( f ) is a subset of the codomain of f . The element y = f ( x ) B is called the value of f at x A or the image of x under f , y is the depenedent variable . x is a pre-image of y under f and is the independent variable or argument of f . We may also consider the action of f on a set, i.e., f ( A ) = Rng( f ). Note: To verify that a given relation f from A to B is a function from A to B , it must be shown that every element of A appears as a first coordinate of exactly one ordered pair in...
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- Spring '08