Basic Structures: Functions, Sequences, and Sums

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Basic Structures Functions, Sequences, and Sums
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Functions

• From calculus, you are familiar with the concept
  of a real-valued function f, which assigns to
  each number x?R a particular value yf(x), where
  y?R.
• But, the notion of a function can also be
  naturally generalized to the concept of assigning
  elements of any set to elements of any set.

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Function Formal Definition

• For any sets A, B, we say that a function f from
  (or mapping) A to B (fA?B) is a particular
  assignment of exactly one element f(x)?B to each
  element x?A.

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Graphical Representations

• Functions can be represented graphically in
  several ways

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Some Function Terminology

• If fA?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 b ?a f(a)b .

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Range versus Codomain

• The range of a function might not be its whole
  codomain.
• The codomain is the set that the function is
  declared to map all domain values into.
• The range is the particular set of values in the
  codomain that the function actually maps elements
  of the domain to.

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Range vs. Codomain - Example

• Suppose I declare to you that f is a function
  mapping students in this class to the set of
  grades A,B,C,D,F.
• At this point, you know fs codomain is
  __________, and its range is ________.
• Suppose the grades turn out all As and Bs.
• Then the range of f is _________, but its
  codomain is __________________.

A,B,C,D,F
A,B,C,D,F
A,B
A,B,C,D,F
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Operators (general definition)

• An n-ary operator over the set S is any function
  from the set of ordered n-tuples of elements of
  S, to S itself.
• E.g., if ST, F, ? can be seen as a unary
  operator, and ?,? are binary operators on S.
• Another example ? and ? are binary operators on
  the set of all sets.

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Constructing Function Operators

• If ? (dot) is any operator over B, then we can
  extend ? to also denote an operator over
  functions fA?B.
• E.g. Given any binary operator ?B?B?B, and
  functions f,gA?B, we define(f ? g)A?B to be
  the function defined by?a?A, (f ? g)(a)
  f(a)?g(a).

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Function Operator Example

• ?, (plus, times) are binary operators over
  R. (Normal addition multiplication.)
• Therefore, we can also add and multiply functions
  f,gR?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)

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Function Composition Operator

• For functions gA?B and fB?C, there is a special
  operator called compose (?).
• It composes (creates) a new function out of f, g
  by applying f to the result of g.
• (f ? g)A?C, where (f ? g)(a) f(g(a)).
• Note g(a)?B, so f(g(a)) is defined and ?C.
• Note that ? (like Cartesian ?, but unlike ,?,?)
  is non-commuting. (Generally, f ? g ? g ? f.)

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Images of Sets under Functions

• Given fA?B, and S?A,
• The image of S under f is simply the set of all
  images (under f) of the elements of S.f(S) ?
  f(s) s?S ? b ? s?S f(s)b.
• Ex Let Aa, b, c, d, e and B1, 2, 3, 4 with
  f(a) 2, f(b) 1, f(c) 4, f(d) 1 and f(e)
  1. The image of the subset S b, c, d is the
  set f(S) 1, 4.

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One-to-One Functions

• A function is one-to-one (1-1), or injective, or
  an injection, iff every element of its range has
  only 1 pre-image.
• Formally given fA?B,x is injective ?
  (??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.

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One-to-One Functions

• Each element of the domain is injected into a
  different element of the range.
• Ex Determine whether the function f from a, b,
  c, d to 1, 2, 3, 4, 5 with f(a)4, f(b)5,
  f(c)1 and f(d)3 is one-to-one.

Yes
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One-to-One Illustration

• Bipartite (2-part) graph representations of
  functions that are (or not) one-to-one

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Sufficient Conditions for 1-1ness

• For functions f over numbers,
• f is strictly (or monotonically) increasing iff
  xgty ? f(x)gtf(y) for all x, y in domain
• f is strictly (or monotonically) decreasing iff
  xgty ? f(x)ltf(y) for all x, y in domain
• If f is either strictly increasing or strictly
  decreasing, then f is one-to-one. e.g. x3

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Onto (Surjective) Functions

• A function fA?B is onto or surjective or a
  surjection iff its range is equal to its codomain
  (?b?B, ?a?A f(a)b).
• An onto function maps the set A onto (over,
  covering) the entirety of the set B, not just
  over a piece of it.
• E.g., for domain codomain R, x3 is onto,
  whereas x2 isnt. (Why not?)

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Illustration of Onto

• Some functions that are or are not onto their
  codomains

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Bijections

• A function f is a one-to-one correspondence, or a
  bijection, or reversible, or invertible, iff it
  is both one-to-one and onto.
• For bijections fA?B, there exists an inverse of
  f, written f ?1B?A, which is the unique function
  such that
• (I is the identity function)

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Inverse Function

• Definition Let f be a one-o-one correspondence
  from the set A to the 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 of f is denoted
  by f-1. Hence, f-1(b)a when f(a)b.

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The Identity Function

• For any domain A, the identity function IA?A
  (variously written, IA, 1, 1A) is the unique
  function such that ?a?A I(a)a.
• Some identity functions youve seen
• ?ing 0, ing by 1, ?ing with T, ?ing with F, ?ing
  with ?, ?ing with U.
• Note that the identity function is both
  one-to-one and onto (bijective).

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Identity Function Illustrations

• The identity function

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Graphs of Functions

• We can represent a function fA?B as a set of
  ordered pairs (a, f(a)) a?A.
• Note that ?a, there is only 1 pair (a, f(a)).
• For functions over numbers, we can represent an
  ordered pair (x, y) as a point on a plane. A
  function is then drawn as a curve (set of points)
  with only one y for each x.

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A Couple of Key Functions

• In discrete math, we will frequently use the
  following functions over real numbers
• ?x? (floor of x) is the largest integer ? x.
• ?x? (ceiling of x) is the smallest integer ? x.

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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.

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Plots with floor/ceiling

• Note that for f(x)?x?, the graph of f includes
  the points (a, 0) for all values of a such that
  a?0 and alt1, but not for a1. We say that the
  set of points (a, 0) that is in f does not
  include its limit or boundary point (a, 1).
• Sets that do not include all of their limit
  points are called open sets. In a plot, we draw
  a limit point of a curve using an open dot
  (circle) if the limit point is not on the curve,
  and with a closed (solid) dot if it is on the
  curve.

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Plots with floor/ceiling Example

• Plot of graph of function f(x) ?x/3?

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Review of Functions

• Function variables f, g, h,
• Notations fA?B, f(a), f(A).
• Terms image, pre-image, domain, codomain, range,
  one-to-one, onto, strictly (in/de)creasing,
  bijective, inverse, composition.
• Function unary operator f ?1, binary operators
  ?, ?, etc., and ?.
• The R?Z functions ?x? and ?x?.

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Sequences Summation

• A sequence or series is just like an ordered
  n-tuple, except
• Each element in the series has an associated
  index number.
• A sequence or series may be infinite.
• A summation is a compact notation for the sum of
  all terms in a (possibly infinite) series.

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Sequences

• Formally A sequence or series an is identified
  with a generating function f S?A for some subset
  S?N (often SN or SN?0) and for some set A.
• If f is a generating function for a series an,
  then for n?S, the symbol an denotes f(n), also
  called term n of the sequence.
• The index of an is n. (Or, often i is used.)

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Sequence Examples

• Many sources just write the sequence a1, a2,
  instead of an, to ensure that the set of
  indices is clear.
• An example of an infinite series
• Consider the series an a1, a2, , where
  (?n?1) an f(n) 1/n. Then an 1, 1/2, 1/3,

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Example with Repetitions

• Consider the sequence bn b0, b1, (note 0 is
  an index) where bn (?1)n.
• bn 1, ?1, 1, ?1,
• Note repetitions! bn denotes an infinite
  sequence of 1s and ?1s, not the 2-element set
  1, ?1.

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Recognizing Sequences

• Sometimes, youre given the first few terms of a
  sequence, and you are asked to find the
  sequences generating function, or a procedure to
  enumerate the sequence.
• Examples Whats the next number?
• 1,2,3,4,
• 1,3,5,7,9,
• 2,3,5,7,11,...

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The Trouble with Recognition

• The problem of finding the generating function
  given just an initial subsequence is not well
  defined.
• This is because there are infinitely many
  computable functions that will generate any given
  initial subsequence.
• We implicitly are supposed to find the simplest
  such function (because this one is assumed to be
  most likely).

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What are Strings, Really?

• We say finite sequences of the form a1, a2, ,
  an are called strings.
• Strings are often restricted to sequences
  composed of symbols drawn from a finite alphabet.
• The length of a (finite) string is its number of
  terms (or of distinct indexes).

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Strings, more formally

• Let ? be a finite set of symbols, i.e. an
  alphabet.
• A string s over alphabet ? is any sequence si
  of symbols, si??, indexed by N or N?0.
• If a, b, c, are symbols, the string s a, b,
  c, can also be written abc (i.e., without
  commas).
• If s is a finite string and t is a string, the
  concatenation of s with t, written st, is the
  string consisting of the symbols in s, in
  sequence, followed by the symbols in t, in
  sequence.

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More String Notation

• The length s of a finite string s is its number
  of positions (i.e., its number of index values
  i).
• If s is a finite string and n?N, sn denotes the
  concatenation of n copies of s.
• ? denotes the empty string, the string of length
  0.
• If ? is an alphabet and n?N,?n ? s s is a
  string over ? of length n, and? ? s s is a
  finite string over ?.

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Summation Notation

• Given a series an, an integer lower bound (or
  limit) j?0, and an integer upper bound k?j, then
  the summation of an from j to k is written and
  defined as follows

• Here, i is called the index of summation.

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Generalized Summations

• For an infinite series, we may write

• To sum a function over all members of a set
  Xx1, x2,

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Simple Summation Example

• An infinite series with a finite sum

• Using a predicate to define a set of elements to
  sum over

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Summation Manipulations

• Some handy identities for summations

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More Summation Manipulations

• Other identities that are sometimes useful

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Example Impress Your Friends

• Boast, Im so smart give me any 2-digit number
  n, and Ill add all the numbers from 1 to n in my
  head in just a few seconds.
• i.e., Evaluate the summation
• There is a simple closed-form formula for the
  result, discovered by Euler at age 12!

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Eulers Trick, Illustrated

• Consider the sum12(n/2)((n/2)1)(n-1)n
• n/2 pairs of elements, each pair summing to n1,
  for a total of (n/2)(n1).

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Example Geometric Progression

• A geometric progression is a series of the form
  a, ar, ar2, ar3, , ark, where a,r?R.
• The sum of such a series is given by

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Some Shortcut Expressions
Geometric series.
Eulers trick.
Quadratic series.
Cubic series.
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Using the Shortcuts

• Example Evaluate
• Use series splitting.
• Solve for desired summation.
• Apply quadratic series rule.

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Nested Summations

• These have the meaning youd expect.
• Note issues of free vs. bound variables, just
  like in quantified expressions, integrals, etc.

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Summations Conclusion

• You need to know
• How to read, write evaluate summation
  expressions like
• Summation manipulation laws we covered.
• Shortcut closed-form formulas, how to use them.