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CS2800-Functions-Seqs_v.5

# CS2800-Functions-Seqs_v.5 - Discrete Math CS 2800 Prof Bart...

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1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Basic Structures: Functions and Sequences Rosen 2.3 and 2.4

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Functions Suppose we have: How do you describe the yellow function ? What’s a function ? f(x) = -(1/2)x – 1/2 x f(x)
Functions More generally: Definition: Given A and B, nonempty sets, a function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a)=b if b is the element of B assigned by function f to the element a of A. If f is a function from A to B, we write f : A B. Note: Functions are also called mappings or transformations. B

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4 Functions A = {Michael, Toby , John , Chris , Brad } B = { Kathy, Carla, Mary} Let f: A B be defined as f(a) = mother(a). Michael Toby John Chris Brad Kathy Carol Mary A B
5 Functions More generally: f: R R, f(x) = -(1/2)x – 1/2 domain co-domain A - Domain of f B- Co-Domain of f B

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6 Functions More formally: a function f : A B is a subset of AxB where 2200 a A, 5 ! b B and <a,b> f. (note: 5 ! for unique exists.) A B A B a point! a collection of points! Why not?
Functions - image & preimage For any set S A, image(S) = {b : 5 a S, f(a) = b} So, image({Michael, Toby}) = {Kathy} image(A) = B - {Carol} image(S) image(John) = {Kathy} pre-image(Kathy) = {John, Toby, Michael} rangeof f image(A) Michael Toby John Chris Brad Kathy Carol Mary A B

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8 Functions - injection A function f: A B is one-to-one ( injective, an injection ) if 2200 a,b,c, (f(a) = b f(c) = b) a = c Not one-to-one Every b B has at most 1 preimage. Michael Toby John Chris Brad Kathy Carol Mary
9 Functions - surjection A function f: A B is onto ( surjective, a surjection ) if 2200 b B, 5 a A f(a) = b Not onto Every b B has at least 1 preimage. Michael Toby John Chris Brad Kathy Carol Mary

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10 Functions – one-to-one-correspondence or bijection A function f: A B is bijective if it is one-to-oneand onto . Anna Mark John Paul Sarah Carol Jo Martha Dawn Eve Every b B has exactly 1 preimage. An important implication of this characteristic: Thepreimage(f -1 ) is a function! They are invertible. Anna Mark John Paul Sarah Carol Jo Martha Dawn Eve
11 Functions: inverse function Definition: Given f, a one-to-one correspondence from set A to set B, the inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b. The inverse function is denoted f -1 . f -1 (b)=a, when f(a)=b. B

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12 Functions - examples Suppose f: R + R + , f(x) = x 2 . Is f one-to-one? Is f onto? Is f bijective? yes yes yes This function is invertible. here
13 Functions - examples Suppose f: R R + , f(x) = x 2 . Is f one-to-one? Is f onto? Is f bijective? no yes no This function is not invertible.

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14 Functions - examples Suppose f: R R, f(x) = x 2 . Is f one-to-one? Is f onto? Is f bijective? no no no

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16 Functions - composition Let f: A B, and g: B C be functions. Then thecomposition of f and g is: (f o g)(x) = f(g(x)) Note: (f o g) cannot be defined unless the range of g is a subset of the domain of f.
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CS2800-Functions-Seqs_v.5 - Discrete Math CS 2800 Prof Bart...

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