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Set and Set Operations

CMPS 2433- Chapter 2 Partially borrowed from Florida State University Some developed by Dr. H. Set and Set Operations. Introduction. A set is a collection of objects The order of the elements does not matter Objects in a set are called elements

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Set and Set Operations

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  1. CMPS 2433- Chapter 2 Partially borrowed from Florida State University Some developed by Dr. H Set and Set Operations

  2. Introduction • A set is a collection of objects • The order of the elements does not matter • Objects in a set are called elements • A well – defined set is a set in which we can determine if an element belongs to that set. • Examples: • The set of all movies in which John Wayne appears is well – defined. • The set of all courses taught by Dr. Halverson • The set of best TV shows of all time is not well – defined. (It is a matter of opinion.)

  3. Notation • Usually denote a set with a capital letter. • Roster notation is the method of describing a set by listing each element of the set. • Example: Let C = set of all even integers less than 12 but greater than zero. • C = {2, 4, 6, 8, 10} = {10, 6, 8, 4, 2} • Example: Let T = set of all courses taught by Dr. Halverson in summer 2014. • T = {CMPS 1013}

  4. More on Notation • Sometimes impossible list all elements of a set. • Z = The set of integers . The dots mean continue on in this pattern forever and ever. • Z = { …-3, -2, -1, 0, 1, 2, 3, …} • Dots can ONLY be used if the pattern is unmistakable • W = {0, 1, 2, 3, …} = the set of whole numbers.

  5. Set – Builder Notation • Set-Builder Notation: specify the rule that determines set membership • First: indicate type of elements in set • Second: specify distinguishing rule • V = { people | citizens registered to vote in Wichita County} • A = {x is a real number | x > 5} • The symbol | is read as “such that”

  6. Special Sets of Numbers • N = The set of natural numbers. = {1, 2, 3, …}. • W = The set of whole numbers. ={0, 1, 2, 3, …} • Z = The set of integers. = { …, -3, -2, -1, 0, 1, 2, 3, …} • Q = The set of rational numbers. ={x| x=p/q, where p and q are elements of Z and q ≠ 0} • H = The set of irrational numbers. • R = The set of real numbers. • C = The set of complex numbers.

  7. Universal Set and Subsets • Universal Set (denoted by U) - set of all possible elements used in a problem • Universal set  Whole numbers (if counting) • Universal set  Rational numbers (if measuring) • Subset: B A if every element of B is also an element A • Example A={1, 2, 3, 4, 5} and B={2, 3} Let S={1,2,3}, list all the subsets of S. • The subsets of S are , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.

  8. The Empty Set • Empty Set: set containing no elements; zero elements • denoted as { } or • Empty Set  of all sets • Do not be confused by this question: • Is this set {0} empty? • It is not empty! It contains one element - zero

  9. Intersection of sets • Intersection of sets A & B is denoted A ∩ B • A ∩ B = {x| x is an element of A and x is an element of B} • A={1, 3, 5, 7, 9} & B={1, 2, 3, 4, 5} • A ∩ B = {1, 3, 5} • B ∩ A = {1, 3, 5}

  10. Union of sets • Union of two sets A, B is denoted A U B and is defined A U B = {x| x is in A or x is in B} • A={1, 3, 5, 7, 9} & B={1, 2, 3, 4, 5} • A U B = {1, 2, 3, 4, 5, 7, 9}. • NOTE: • Order of listing sets does not matter • Never repeat elements

  11. Difference of Sets • Difference of sets A & B denoted A-B • A-B = {x|x is in A but x is NOT in B} • Like subtraction • A={1, 3, 5, 7, 9} & B={1, 2, 3, 4, 5} • A-B = {7, 9} • B-A = {2, 4}

  12. Mutually Exclusive Sets • Two sets A and B are Disjoint (aka Mutually Exclusive) if A ∩ B = • I.E. No elements in common • Think of this as two events that can not happen at the same time.

  13. Complement of a Set • Complement of set A is denoted by Ᾱor by Ac • Ᾱ = {x | x is in the universal set but x is not in set A} • U={1,2,3,4,5} & A={1,2}, • Ᾱ = {3,4,5}

  14. Cardinal Number • Cardinalityof a set is the number of elements in the set and is denoted by |A| or n(A) • A={2,4,6,8,10}, then |A|=5. • Cardinality formula • |A U B|=|A| + |B| – |A∩B| • |{ } | = ??? (Cardinality of the empty set?)

  15. Theorem 2.1 (pg. 43) • Commutative Law • A U B = B U A and B ∩ A = A ∩ B • Associative Law • (A U B) U C = A U (B U C) • (A ∩ B) ∩ C = A ∩ (B ∩ C) • Distributive Law • A U (B ∩ C) = (A U B) ∩ (A U C) • A U(B ∩ C) = (A ∩ B) U (A∩ C) • See others on Page 43

  16. Ordered Pair • An Ordered Pair of elements a & b is denoted (a,b) • Order is significant • (a,b) ≠ (b,a)

  17. Cartesian Product • Cartesian Product of sets A & B • Denoted A X B • is the set consisting of all ordered pairs (a,b) where a is an element of A and b is an element of B • A X B = { (a,b)| a is in A & b is in B} • If |A| = 3 and |B| = 7, what is |A X B|? • Can you list them?

  18. De Morgan’s Laws (pg. 45) • See Page 45 – Memorize • Study Proofs also! • And YOU prove the part not proven in the book

  19. Homework – 2.1 • Pages 46 & 47 • 1-8, 13-28

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