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2.1 Sets

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Section 2.1
Section Summary
 Definition of sets
 Describing Sets
 Roster Method
 Set-Builder Notation
 Some Important Sets in Mathematics
 Empty Set and Universal Set
 Subsets and Set Equality
 Cardinality of Sets
 Tuples
 Cartesian Product
Sets
 A set is collection of objects.
 the students in this class
 the chairs in this room
 The contents have to be clearly determined
 The objects in a set are called the elements, or members of
the set. A set is said to contain its elements.
 Set notation uses { }
 The notation a ∈ A denotes that a is an element of the set
A.
 If a is not a member of A, write a ∉ A
Describing a Set: Roster Method
 Sets are named with capital letters
 Roster method is listing all elements
 S = {a,b,c,d}
 Order not important
S = {a,b,c,d} = {b,c,a,d}
 Each distinct object is either a member or not; listing more
than once does not change the set.
S = {a,b,c,d} = {a,b,c,b,c,d}
 Elipses (…) may be used to describe a set without listing all
of the members when the pattern is clear.
S = {a,b,c,d, …,z }
Roster Method
 Set of all vowels in the English alphabet:
V = {a,e,i,o,u}
 Set of all odd positive integers less than 10:
O = {1,3,5,7,9}
 Set of all positive integers less than 100:
S = *1,2,3,…,99+
 Set of all integers less than 0:
S = *…., -3,-2,-1}
Tedious or Impossible
 Roster method is sometime tedious or impossible
 Example: (tedious)
 L={x | x is a lowercase letter of the alphabet}
 L= {a, b, c … z}
 Example: (impossible)
 Z = {x | x is a positive number}
 Z = {1, 2, 3, 4 …}

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2.1 Sets

  • 2. Section Summary  Definition of sets  Describing Sets  Roster Method  Set-Builder Notation  Some Important Sets in Mathematics  Empty Set and Universal Set  Subsets and Set Equality  Cardinality of Sets  Tuples  Cartesian Product
  • 3. Sets  A set is collection of objects.  the students in this class  the chairs in this room  The contents have to be clearly determined  The objects in a set are called the elements, or members of the set. A set is said to contain its elements.  Set notation uses { }  The notation a ∈ A denotes that a is an element of the set A.  If a is not a member of A, write a ∉ A
  • 4. Describing a Set: Roster Method  Sets are named with capital letters  Roster method is listing all elements  S = {a,b,c,d}  Order not important S = {a,b,c,d} = {b,c,a,d}  Each distinct object is either a member or not; listing more than once does not change the set. S = {a,b,c,d} = {a,b,c,b,c,d}  Elipses (…) may be used to describe a set without listing all of the members when the pattern is clear. S = {a,b,c,d, …,z }
  • 5. Roster Method  Set of all vowels in the English alphabet: V = {a,e,i,o,u}  Set of all odd positive integers less than 10: O = {1,3,5,7,9}  Set of all positive integers less than 100: S = *1,2,3,…,99+  Set of all integers less than 0: S = *…., -3,-2,-1}
  • 6. Tedious or Impossible  Roster method is sometime tedious or impossible  Example: (tedious)  L={x | x is a lowercase letter of the alphabet}  L= {a, b, c … z}  Example: (impossible)  Z = {x | x is a positive number}  Z = {1, 2, 3, 4 …}
  • 7. Some Important Sets N = natural numbers = {0,1,2,3….+ Z = integers = *…,-3,-2,-1,0,1,2,3,…+ Z⁺ = positive integers = {1,2,3,…..+ R = set of real numbers R+ = set of positive real numbers C = set of complex numbers. Q = set of rational numbers
  • 8. Set-Builder Notation  Specify the property or properties that all members must satisfy: S = {x | x is a positive integer less than 100} In word: The set S is all the elements x such that x is positive and less than 100 O = {x | x is an odd positive integer less than 10} O = {x ∈ Z⁺ | x is odd and x < 10}  Positive rational numbers: Q+ = {x ∈ R | x = p/q, for some positive integers p,q} Such thatAll Elements x
  • 9. Set-Builder Notation Ex: Express Set A={x | x is a month begins with the letter M} using the roster method. Word Description Roster Method Set-Builder Notation W is the set of all days of the week W= {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} W={x | x is a day of the week}
  • 10. Some things to remember  Sets can be elements of sets. {{1,2,3},a, {b,c}} {N,Z,Q,R}  The empty set is different from a set containing the empty set. ∅ ≠ * ∅ + { }
  • 11.  Example: Recognize the empty set: 1. {0} No 2. 0 No, it is not a set 3. {x | x is a number less than 4 and greater than 10} No 4. {x | x is a square with 3 sides} No
  • 12. Notations for Set Membersship  ∈  Is an element of  ∈  Is not an element of  Example: Determine if each statement is true or false.  r ∈ *a, b, c … z+  True  7 ∈ *1, 2, 3, 4, 5+  False  *a+ ∈ *a, b+  False
  • 13. Set Equality Definition: Two sets are equal if and only if they have the same elements.  Therefore if A and B are sets, then A and B are equal if and only if .  We write A = B if A and B are equal sets. {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5}
  • 14. Subsets Definition: The set A is a subset of B, if and only if every element of A is also an element of B.  The notation A ⊆ B is used to indicate that A is a subset of the set B.  A ⊆ B holds if and only if is true.
  • 15. Showing a Set is or is not a Subset of Another Set  Showing that A is a Subset of B: To show that A ⊆ B, show that if x belongs to A, then x also belongs to B.  Showing that A is not a Subset of B: To show that A is not a subset of B, A ⊈ B, find an element x ∈ A with x ∉ B. (Such an x is a counterexample to the claim that x ∈ A implies x ∈ B.) Examples: The set of all computer science majors at your school is a subset of all students at your school.
  • 16. Another look at Equality of Sets  Recall that two sets A and B are equal, denoted by A = B, iff  Using logical equivalences we have that A = B iff  This is equivalent to A ⊆ B and B ⊆ A
  • 17. Proper Subsets Definition: If A ⊆ B, but A ≠B, then we say A is a proper subset of B, denoted by A ⊂ B. If A ⊂ B, then is true.
  • 18. Set Cardinality Definition: If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite. Definition: The cardinality of a finite set A, denoted by |A|, is the number of elements of A. Examples: 1. |ø| = 0 2. Let S be the letters of the English alphabet Then |S| = 26 3. C= {1,2,3} |C|=3 4. D= {1, 1, 2, 3, 3} |D|= 3 5. E={ø} |E|= 1 The set of integers is infinite. Finite: ends Infinite: no end
  • 19. Universal Set and Venn Diagram  The universal set U: is the set containing everything currently under consideration.  Contents depend on the context.  Venn Diagram:  visual relationship among sets  Uses circles and rectangles U Venn Diagram a e i o u V John Venn (1834-1923) Cambridge, UK
  • 20. Example Use the Venn Diagram to determine each of the following sets: a) U U={$, 5, s, w} b) B B={3} U B 5 $ 3 s w
  • 21. Representing 2 sets in a Venn Diagram  4 different ways U A Disjoint Set Proper Subset U AB B U A=B Equal Sets Set with some common elementsU A B
  • 22. Example Use the Venn Diagram to determine each of the following sets: a) U U={a, b, c, d, e, f} b) B B={c, d} c) The set of elements in A but not B {a, b} d) The set of elements in U but not B {a, b, e, f} e) The set of elements both in A and B {c} U B c b a f e A d
  • 23. Example Use the Venn Diagram to illustrate the relationship: A ⊆ B and B ⊆ C U BA C
  • 24. Power Sets Definition: The set of all subsets of a set A, denoted P(A), is called the power set of A. Example: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}}  If a set has n elements, then the cardinality of the power set is 2ⁿ.
  • 25. Cartesian Product Definition: The Cartesian Product of two sets A and B, denoted by A × B is the set of ordered pairs (a,b) where a ∈ A and b ∈ B . Example: A = {a,b} B = {1,2,3} A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)}  Definition: A subset R of the Cartesian product A × B is called a relation from the set A to the set B. René Descartes (1596-1650)
  • 26. Cartesian Product Example: What is A × B × C where A = {0,1}, B = {1,2} and C = {0,1,2} Solution: A × B × C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1), (0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,1,2)}
  • 27. Let A ={a, b, c}, B= {x, y}, and C={0, 1} Find: a) A × B × C {(a, x, 0), (a, x, 1), (a, y, 0), (a, y, 1), (b, x, 0), (b, x, 1), (b, y, 0), (b, y, 1), (c, x, 0), (c, x, 1), (c, y, 0), (c, y, 1)} b) C × B × A {(0, x, a), (0, x, b), (0, x, c), (0, y, a), (0, y, b), (0, y, c), (1, x, a), (1, x, b), (1, x, c), (1, y, a), (1, y, b), (1, y, c)}