Chapter 3

• Four Basic Concepts


  • Set
  • Relation
  • Function
  • Binary Operation
• Set


  A well-defined collection of distinct objects
• Element or member of a set


  An object belonging to a set
• Notation "∈"


  Indicates that a specific element belongs to a set
• Notation "∉"


  Indicates that a specific element does not belong to a set
• Set C of counting numbers less than 4

  • 1
  • 2
  • 3
• Roster method


  Enumerating the elements of a set, separated by commas and enclosed in braces
• Rule method


  Using a phrase to describe all the elements in the set
• Empty Set or Null Set

  A set with no elements, denoted by ∅ or { }
• Singleton Set


  A set with only one element
• Universal Set


  A set that contains all the elements under consideration, denoted by U
• Finite Set


  A set that consists of a finite number of elements
• Infinite Set


  A set that consists of an infinite number of elements
• Equal Sets


  Two sets A and B are equal if they have exactly the same elements, written as A=B
• Equivalent Sets


  Two sets A and B have the same number of elements
• Subset


  A set A is a subset of a set B, written as A⊆B, if every element of A is also an element of B
• Proper Subset


  A set A is a proper subset of a set B, written as A⊂B, if A⊆B and A≠B
• The null set is a proper subset of every set
• Any set is a subset of itself
• A set with n elements has a total of 2^n subsets
• Subsets of A = {1, 3}


  • {}
  • {1}
  • {3}
  • {1, 3}
• Union of sets A and B


  The set containing all the elements which belong to either A or B or to both, written as A∪B
• Union of sets A and B

  • A = {a, b}, B = {a, x, w}, U = {a, b, c, w, x, y}

  A∪B = {a, b, x, w}
• Intersection of sets A and B

  The set of all elements which are common to both A and B, written as A∩B
• Intersection of sets A and B


  • A = {a, b}, B = {a, x, w}, U = {a, b, c, w, x, y}

  A∩B = {a}
• Complement of set A

  The set of all elements which are in the universal set U but not in A, written as A'
• Complement of set A


  • A = {a, b}, U = {a, b, c, w, x, y}

  A' = {c, w, x, y}
• Cartesian product of sets A and B


  The set of all ordered pairs (a,b) where a∈A and b∈B, written as A×B
• Cartesian product of sets A and B

  • A = {1, 5}, B = {2, 3, 5}

  A×B = {(1,2), (1,3), (1,5), (5,2), (5,3), (5,5)}
• Relation


  A relationship between sets of information
• Relation R from set X to set Y


  A subset of X×Y
• Domain of a relation R


  The set of all the first coordinates in the ordered pairs in R
• Image of a relation R


  The set of all the second coordinates in the ordered pairs in R
• Function


  A relation from A to B where for each a∈A, there corresponds exactly one b∈B
• Representation of functions


  Mapping, Graph, Ordered pairs, Equations
• Vertical Line Test


  A graph represents a function if there are no vertical lines that intersect the graph at more than one point
• Binary Operation


  A function from S×S into S such that for x,y∈S, we have x*y
• Commutative Property


  A binary operation * on S is commutative if for all x,y∈S, x*y = y*x
• Associative Property


  A binary operation * on S is associative if for all x,y,z∈S, (x*y)*z = x*(y*z)
• Addition (+), subtraction (-) and multiplication (·) are binary operations on the set of real numbers ℝ