Basic Structures: Sets, Functions, Sequences, and Sums

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Chapter 2

• Basic Structures Sets, Functions, Sequences, and
  Sums

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Sec 2.1

• Sets

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Definitions Set element of contains Ø

• The objects in the set are called elements or
  members of the set.
• A set is said to contain its elements.
• The empty set Ø is a set which contains no
  elements.
• The universal set, U is the set of all elementss
  under consideration

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Standard Sets

• N - Set of natural numbers,0,1,2,3,
• Z - Set of integers,,-2,-1,0,1,2,
• Z - Set of positive integers,1,2,3,
• Q - Set of rational numbers,p/qp?Z,q?Z,q?0
• R - Set of real numbers

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Definitions Set equality Subset Finite and
Infinite Cardinality

• Two sets are equal if they have the same
  elements.
• Set A is said to be a subset of set B (A ? B) if
  every element of A is also an element of set B.
• Set A is a proper subset of set B (A ? B) if A ?B
  and A ? B.
• A set with n distinct elements is said to be a
  finite set.
• S - The cardinality of set S is the number n
  of elements is the set.
• A set that is not finite is called infinite

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Definitions Power Set, Ordered n-tuple
Cartesian Product

• P(S) - The power set of set S is the set of all
  subsets of S.
• The ordered n-tuple (a1, a2, , an) is the
  ordered collection where for each i, ai ? Ai.
• A x B The Cartesian product of sets A and B is
  the set of all ordered pairs (a,b) with a ? A and
  b ? B.
• A1 x A2 X x An - The Cartesian product of sets
  A1, A2, , An is the set of all ordered
  n-tuples (a1, a2, , an) where ai ? Ai.
• ?

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Homework

• Sec 2.1
• pg. 119 1,3,5,7,13,17,21, 23, 28, 29

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Sec 2.2

• Set Operations

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Definitions Union Intersection Empty set
Disjoint Sets

• A?B The union of sets A and B is the set of all
  elements that are contained in either A or B or
  both.
• A?B - The intersection of sets A and B is the set
  of all elements that are contained in both A and
  B.
• ? - The empty set is the set with no elements.
• Disjoint Two sets are disjoint if the
  intersection of these two sets is the empty set.
• ?

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Definitions Universal Set Complement, Difference

• Universal set, U The universal set contains all
  elements under consideration.
• A The complement of set A is the set of all
  elements in the Universal set that are not in A.
• B A The difference of B and A (or the
  complement of A relative to B) is the set of all
  elements in B, except those in A, or equivalently
  B ? A.
• ?

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Set Properties

• Identity Law A ? ? A, A ? U A
• Domination Law A ? U U, A ? ? ?
• Idempotent Laws A ? A A, A ? A A
• Complementation Laws (Ac)c A Note here Ac A
• Commutative Laws A ? B B ? A, A ? B B ?
  A
• Associative Laws A ? (B ? C) (A ? B) ? C,
  A ? (B ? C) (A ? B) ? C
• Distributive Laws A ? (B ? C) (A ? B) ? (A ?
  C) A ? (B ? C) (A ? B) ? (A ? C)
• De Morgans Laws (A ? E)c A ? E, (A ? E)c A
  ? E
• Absorption Laws A ? (A ? B) A, A ? (A ? B)
  A
• Complement Laws A ? A U, A ? A ?

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Proving Set Identities

• To prove that two sets A and B are equal
• Method I Show that A ? B and B ? A. That is
  take an element x from A and, using logic, verify
  that x is in B and conversely argue that if x?B
  then x?A.
• Method II If set A and B are formed by
  combining sets, use a set membership table to
  show that sets A and B have identical columns in
  the table.

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Generalized Unions Intersections

• Generalized union of a collection of sets is the
  sets that contains those elements that are
  members of at least one set in the collection.
• Notation
• Generalized intersection of a collection of sets
  is the sets that contains those elements that are
  members of all the sets in the collection.
• Notation

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Computer Representation of Sets

• Universal set U with the bit string of length n
• a1,a2,,an
• Subset A of U is represented a bit string with 1
  if ai belongs to A 0 if ai not
• eg U1,2,3,4,5,6, A1,3,5, A is 101010
• Boolean Operations
• 1?11 1?00
• 1?11 1?01
• eg 101 ? 011001 101 ? 011 111

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Homework

• Sec 2.2
• pg. 130 1,3,11,13,15,17,23,29,49(a,b), 50, 51

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Section 2.3

• Functions

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Definition function

• f A ? B A function, f, is a correspondence
  between two sets, A and B, such that for each
  element of set A there corresponds exactly one
  element of the B.
• Notation f(a) b denotes the fact that the
  function makes the assignment between the value
  a ? A and the value b ?B
• If f A ? B then we say f maps A to B.
• ?

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Definitions Domain Codomain Image Pre-image
Range Maps

• The domain of the function fA?B is the set A.
• The codomain is the set B.
• If f(a) b, then b is called the image of a and
  a is called the pre-image of b.
• If S is a subset of A, the image of S is the
  subset of B that contains all the images of
  elements of S.
• The Range of f is the set of all images of
  elements of A.
• The function fA?B is said to map set A to set B.
• ?

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Increasing/Decreasing Functions

• Definition Let f be a function whose domain and
  codomain are subsets of the real numbers and
  suppose x and y are in the domain of f.
• f is said to be increasing if f(x) ? f(y)
  whenever x lt y.
• f is said to be strictly increasing if f(x) lt
  f(y) whenever x lt y.
• f is said to be decreasing if f(x) ? f(y)
  whenever x lt y.
• f is said to be strictly decreasing if f(x) gt
  f(y) whenever x lt y.

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Definitions Injection, Surjection, Bijection

• Let f A ? B be a function.
• f is injective or one-to-one iff f(x) f(y)
  implies that x y for all x and y in A.
• f is surjective or onto if set B is the image of
  A (i.e. ?b ?B ? a?A such f(a) b.)
• f is a bijection or a one-to-one correspondence
  if f is both surjective and injective (i.e. it is
  both one-to-one and onto)
• ?

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Definitions Inverse

• Let fA?B be a one-to-one correspondence from set
  A to set B. The inverse of f is the function
  that assigns to each b?B the unique element a?A
  such that f(a) b. The inverse function is
  denoted by f-1.
• Note f-1(b) a if and only if f(a) b, thus
• ?y?B f(f-1(b)) b and ?x?A f-1(f(a)) a.
• ?

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Definition Composition of two functions, Graphs

• Let g A ? B and f B ? C be two functions. The
  composition of f and g denoted by f o g is
  defined by
• f o g (x) f(g(x)). The domain of the function
  is the set of x in the domain of f such that g(x)
  is in the domain of f.
• The graph of a function f A?B is the set of all
  ordered pairs (a,b) a?A and f(a)b
• ?

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Graph, ceiling, floor

• Let fA ? B. The graph of the function f is the
  set of ordered pairs (a,b)a?A bf(a)
• Ceiling f(x) x The ceiling function assigns
  to the real number x the smallest integer that is
  greater than or equal to x.
• Floor f(x) x The floor function assigns to
  the real number x the largest integer that is
  less than or equal to x.
• Factorial Function fN ?Z, f(n)n!123n
• Define f(0)0!1

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Useful results for Floor Ceiling Function

• 1a. xn iff n ?xltn1
• 1b. xn iff x-1ltn ?x
• 1c. xn iff n-1ltx ?n
• 1d. xn iff x ?nltx1
• 2. x-1lt x ? x? x ltx1
• 3a. -x -x
• 3b. -x - x
• 4a. xn xn
• 4b. xn x n

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Homework

• Sec 2.3
• pg. 133 1,9,10,11,19,23,27,32,33

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Sec 2.4

• Sequences and Summation

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Definitions

• Sequence an A sequence is a function whose
  domain is either the set 0, 1, 2, or the set
  1, 2, 3, and whose codomain is a generally a
  set of numbers. We use the notation an to denote
  the image of the integer n and we call an a term
  of the sequence.
• Geometric Progression A geometric progression
  is a sequence of the form a, ar, ar2, , arn.
  The number a is called the initial term and the
  number r is called the common ratio.
• Arithmetic Progression An arithmetic
  progression is a sequence of the form a, ad,
  a2d, , and. The number a is called the
  initial term and the number d is called the
  common difference.
• ?

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Summation Notation
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Geometric Progression Theorem
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Useful Summation Results
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Definitions

• Cardinality The sets A and B have the same
  cardinality if and only if there is a one-to-one
  correspondence from A to B.
• A set that is either finite or has the same
  cardinality as the set of positive integers is
  called countable. A set that is not countable is
  called uncountable.
• ?

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Theorems

• Theorem The set of all Rational numbers is
  countable
• Theorem The set of all real numbers in the
  interval 0,1 are not countable.

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Homework

• Sec 2.4
• pg. 161 3, 5, 7, 13, 15, 17, 31

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THE END

• Chapter 2