Set and Set Operations

CMPS 2433- Chapter 2Partially 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 Z = The set of integers . The dots mean

continue on in this pattern forever and ever.• Z Z = { …-3, -2, -1, 0, 1, 2, 3, …}• Dots can ONLY be used if the pattern is

unmistakable• WW = {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• NN = The set of natural numbers.

= {1, 2, 3, …}.• WW = The set of whole numbers. ={0, 1, 2, 3, …}• ZZ = The set of integers.

= { …, -3, -2, -1, 0, 1, 2, 3, …}• QQ = The set of rational numbers. ={x| x=p/q, where p and q are elements of Z and Z and q q ≠ 0≠ 0 }• HH = The set of irrational numbers.• RR = 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 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• 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

• 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