Functions and Sequences Rosen, Secs. 2.3, 2.4

Transcription

1 UC Davis, ECS20, Winter 2017 Discrete Mathematics for Computer Science Prof. Raissa D Souza (slides adopted from Michael Frank and Haluk Bingöl) Lecture 8 Functions and Sequences Rosen, Secs. 2.3, 2.4

2 Function: Formal DeOinition Def. For any sets A, B, we say that a func'on f (or mapping ) from A to B is a par3cular assignment of exactly one element f(x) B to each element x A. We can write f:a B For a A and b B, we can evaluate f(a)=b.

3 Important Function Terminology Def. Let f:a B, and f(a)=b (where a A & b B). Then 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 a f(a)=b }.

4 Graphical Representations Func3ons can be represented graphically in several ways: f a b A B Like Venn diagrams f A B Bipartite Graph y (This has f:r -> R) x Plot Note: EVERY element of set A has to be mapped to ONE (and only one) element in B.

5 One-to-One or Injective functions Bipar3te (2-part) graph representa3ons of func3ons that are (or are not) one-to-one: One-to-one Not one-to-one ( Many-to-one instead) Not a function! Note domain codomain for 1-to-1

6 Onto or surjective functions Some func3ons that are, or are not, onto their codomains: Onto (but not 1-1) Not Onto (or 1-1) Both 1-1 and onto 1-1 but not onto Note domain codomain for onto

7 1-1 and onto = bijection/invertible Some func3ons that are, or are not, onto their codomains: Onto (but not 1-1) Not Onto (or 1-1) Both 1-1 and onto 1-1 but not onto

8 Bijections Def. A func3on f is said to be a bijec'on, (or a one-to-one correspondence, or reversible, or inver'ble,) iff it is both one-to-one and onto. Def. For bijec3ons f:a B, there exists an inverse of f, wriqen f -1 :B A, which is the unique func3on such that (where I A is the iden3ty func3on on A) Both 1-1 and onto f 1! f = I A Domain = Codomain = Range We can invert the function!!

9 Graphs of Functions We can represent a func3on f:a B as a set of ordered pairs {(a,f(a)) a A}. Note that a, there is only 1 pair (a,b). Later (ch.6): rela'ons loosen this restric3on. For func3ons over numbers, we can represent an ordered pair (x,y) as a point on a plane. A func3on is then drawn as a curve (set of points), with only one y for each x.

10 A Couple of Key Functions In discrete math, we will frequently use the following two func3ons over real numbers: Def. The floor func3on :R Z, where x ( floor of x ) means the largest (most posi3ve) integer x. Formally, x : max({ j Z j x}). Def. The ceiling func3on :R Z, where x ( ceiling of x ) means the smallest (most nega3ve) integer x. Formally, x : min({ j Z j x})

11 Visualizing Floor & Ceiling Real numbers fall to their floor or rise to their ceiling. Note that if x Z, -x - x & -x - x Note that if x Z, x = x = x =2 1.6 =1-1.4 = = -2-3 = -3 = -3

12 Plots with Oloor/ceiling: Example Plot of graph of func3on f(x) = x/3 : f(x) Set of points (x, f(x)) x Note, the open dot denotes that in the limit the point is not on the curve, and the closed dot means it is on the curve.

13 Operators (general deoinition) Def. An n-ary operator over (or on) the set S is any func3on from the set of ordered n- tuples of elements of S, to S itself. Ex. If S={T,F}, can be seen as a unary operator, and, are binary operators on S. Ex. and are binary operators on the set of all sets. (See HW3 with i k nota3on)

14 Combining Function Operators +, ( plus, 3mes ) are binary operators over R. (Normal addi3on & mul3plica3on.) Therefore, we can also add and mul3ply func'ons Def. Let f, g: R R. (f + g): R R, where (f + g)(x) = f(x) + g(x) (f g): R R, where (f g)(x) = f(x) g(x)

15 Function Composition Operator Def. Let g:a B and f:b C. The composi'on of f and g, denoted by f g, is defined by (f g)=f(g(a)). A g B f C a g(a) f(g(a)) Remark. (like Cartesian, but unlike +,, ) is non-commu3ng. (Generally, f g g f.)

16 Review of 2.3 (Functions) Func3on variables f, g, h, Nota3ons: f:a B, f(a), f(a). Terms: image, preimage, domain, codomain, range, one-to-one, onto, strictly (in/de)creasing, bijec3ve, inverse, composi3on. Func3on unary operator f -1, binary operators +, -, etc., and. The R Z func3ons x and x.

17 2.4: Sequences, Strings, & Summations A sequence or series is just like an ordered n- tuple, except: Each element in the series has an associated index number. E.g., (a 1,a 2,, a k ) A sequence or series may be infinite. A string is a sequence of symbols from some finite alphabet. (e.g., words in a language) A summa'on is a compact nota3on for the sum of all terms in a (possibly infinite) series. k i : a j + a j i= j a + a k

18 Sequences Def. A sequence or series {a n } is iden3fied with a genera'ng func'on f:s A for some subset S N and for some set A. Oien we have S= N or S=Z + = N -{0}.

19 Recognizing Sequences Some3mes, you re given the first few terms of a sequence and you need to find the sequence s genera3ng func3on, which is a procedure to enumerate the sequence. Examples: What s the next number? 1,2,3,4, 1,3,5,7,9, 2,3,5,7,11,... 5 (the 5th smallest number >0) 11 (the 6th smallest odd number >0) 13 (the 6th smallest prime number)

20 Sequence elements, a n Def. If f is a generating function for a series {a n }, then for n S, the symbol a n denotes f(n), also called term n of the sequence. The index of a n is n. (Or, often i is used.) A series is sometimes denoted by listing its first and/or last few elements, and using ellipsis ( ) notation. E.g., {a n } = 0, 1, 4, 9, 16, 25, is taken to mean n N, a n = n 2.

21 Sequence Examples Some authors write the sequence a 1, a 2, instead of {a n }, to ensure that the set of indices is clear. Be careful: oien nota3on leaves the indices ambiguous, but context makes it clear. Ex. An example of an infinite series: Consider the series {a n } = a 1, a 2,, where ( n 1) a n = f(n) = 1/n. Then, we have {a n } = 1, 1/2, 1/3,

22 Switch to blackboard mode