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

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.)

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}

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.

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”

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.

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}.

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

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}

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

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}

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.

Complement of a Set Complement of set A is denoted by Ᾱ or 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}

Cardinal Number |{ } | = ??? (Cardinality of the empty set?) Cardinality of 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?)

Theorem 2.1 (pg. 43) Commutative Law Associative Law Distributive 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

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

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?

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

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