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ics141-lecture09-Functions

ics141-lecture09-Functions - University of Hawaii ICS141...

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9-1 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ICS141: Discrete Mathematics for Computer Science I Department of Information and Computer Sciences University of Hawaii Stephen Y. Itoga
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9-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 9 Chapter 2. Basic Structures 2.3 Functions 2.4 Sequences and Summations
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9-3 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Some material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of “Discrete Mathematics and Its Applications” written by Kenneth H. Rosen. Some material is from Prof. Baek
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9-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Review Previously… Function variables f , g , h , … Notations: f : A B, f ( a ), f ( A ). Terms: image, preimage, domain, codomain, range, one-to-one, composition. Function binary operators + , - , etc. , and ◦. Today Terms: one-to-one, onto, strictly (in/de)creasing, bijective, inverse (unary operator f - 1 ) The R Z functions x and x .
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9-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Review: Function Terminology If it is written that f : A B , and f ( a )= b (where a A & b B ), then we say: A is the domain of f B is the codomain of f b is the image of a under f a is a pre-image of b under f In general, b may have more than 1 pre- image The range R B of f is R ={ b | 5 a f ( a )= b }
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9-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii One-to-One Functions A function f is one-to-one ( 1-1 ), or injective , or an injection , iff f ( a )= f ( b ) implies that a = b for all a and b in the domain of f (i.e. every element of its range has only 1 pre-image). Formally: given f : A B, x is injective” = ( ¬5 x , y : x y f ( x ) = f ( y ) ). Only one element of the domain is mapped to any given one element of the range. Domain & range have same cardinality. What about codomain?
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9-7 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii One-to-One Illustration Bipartite (2-part) graph representations of functions that are (or not) one-to-one: One-to-one Not one-to-one Not even a function! Example 8 : Is the function f :{a, b, c, d} {1, 2, 3, 4, 5} with f (a)=4, f (b)=5, f (c)=1, and f (d)=3 one-to- one? Example 9 : Let f : Z Z be such that f ( x ) = x 2 . Is f one-to-one?
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9-8 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Sufficient Conditions for 1-1ness For functions f over numbers, we say: f is strictly (or monotonically ) increasing iff x>y f ( x ) >f ( y ) for all x,y in domain; f is strictly (or monotonically )
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