ics141-lecture09-Functions

ics141-lecture09-Functions - 9-1ICS 141: Discrete...

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Unformatted text preview: 9-1ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiICS141: Discrete Mathematics for Computer Science IDepartment of Information and Computer SciencesUniversity of HawaiiStephen Y. Itoga9-2ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiLecture 9Chapter 2. Basic Structures2.3 Functions2.4 Sequences and Summations9-3ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSome 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. Baek9-4ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiReviewPreviouslyFunction variables f, g, h, Notations: f:AB, f(a), f(A).Terms: image, preimage, domain, codomain, range, one-to-one, composition.Function binary operators +, -, etc., and .TodayTerms: one-to-one, onto, strictly (in/de)creasing, bijective, inverse (unary operator f -1)The RZfunctions xand x.9-5ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiReview: Function TerminologyIf it is written that f:AB, and f(a)=b (where aA& bB), then we say:Ais the domainof fBis the codomainof fbis the imageof a under fais a pre-imageof bunder fIn general, bmay have more than 1 pre-imageThe rangeRBof f is R={b| 5af(a)=b}9-6ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiOne-to-One FunctionsA function fis one-to-one(1-1), or injective, or aninjection, iff f(a)=f(b) implies that a= bfor all aand bin the domain of f(i.e. every element of its range has only1 pre-image).Formally: given f:AB,x is injective = (5x,y: xy f(x) =f(y)).Only oneelement of the domain is mapped toany given oneelement of the range.Domain & range have same cardinality. What about codomain?9-7ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiOne-to-One IllustrationBipartite (2-part) graph representations of functions that are (or not) one-to-one:One-to-oneNot one-to-oneNot 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 :ZZbe such that f(x) = x2. Is fone-to-one?9-8ICS 141: Discrete Mathematics I (Spr 2008)University of HawaiiSufficient Conditions for 1-1nessFor functions fover numbers, we say:fis strictly(or monotonically) increasingiff x>y f(x)>f(y)for all x,yin domain;fis strictly(or monotonically) decreasingiff x>y f(x)<f(y)for all x,yin domain;If fis either strictly increasing or strictly decreasing, then fis one-to-one. 9-9ICS 141: Discrete Mathematics I (Spr 2008)...
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This note was uploaded on 09/12/2008 for the course ICS 141 taught by Professor Idk during the Fall '08 term at Hawaii.

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ics141-lecture09-Functions - 9-1ICS 141: Discrete...

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