Sets & Set OperationsTuples & Relations

Sets• Sets are collections

– The things in the collection are called elements or members– Sets have no duplicates– Elements in a set have no order

• Special sets– Universe: all elements under consideration (denoted U or E)– Empty set: set with no elements (denoted { } or )– Others: NN, ZZ, RR, etc.

• Notation { }– Enumeration, e.g., {1, 2, 3}, {1, 2, …}, {1, 2, … , 100}, etc.– Set building: { x | P(x) } all elements in E that satisfy predicate P

(e.g., { x | x>5 x<10 } = {6, 7, 8, 9} when E = NN

• Element of: x A

• Cardinality: |A| or #A– Size or number of elements in A, e.g., |{a,b}| = 2

Set Equality, Subsets, Supersets• Set Equality

– A = B iff A and B have the same elements

– A = B xA xB

• Subsets– A B xA xB (subset or equal)– A B A B x(xB xA) (proper subset)

• SupersetsA B iff B A

A B iff B A

Proofs about Set Equalityand the Empty Set

• Prove: A = B iff A B B AA = B xA xB definition of set equality

(xA xB) (xB xA) P Q (P Q) (Q P) A B B A definition of subset

• Prove: A (i.e., is a subset of every set) A x xA definition of subset

F xA x is false (for if not there is an element of E in the empty set, contrary to the definition)

T

Set Operations: Intersection

IntersectionA B {x | xA xB}{1, 2, 3} {2, 3, 4} = {2, 3}

Prove: A B ABy definition, A B A xAB xA

1. xA negate conclusion2. xAB premise3. xA xB def of 4. xA 3, simplification5. xA xA 1&4, conjunction6. F 5, contradiction

Set Operations: Intersection

IntersectionA B {x | xA xB}{1, 2, 3} {2, 3, 4} = {2, 3}

Prove: A B ABy definition, A B A xAB xA

1. xA negate conclusion2. xAB premise3. xA xB def of 4. xA 3, simplification5. xA xA 1&4, conjunction6. F 5, contradiction

A simpler (direct) proof

A B

Set Operations: Union

UnionA B {x | xA xB}{1, 2, 3} {2, 3, 4} = {1, 2, 3, 4}No duplicates!

Prove: A A BBy definition, A AB xA xA xB

1. xA premise2. xA xB 1, law of addition

A B

Set Operations: Set Difference

Difference (minus)A – B {x | xA xB}{1, 2, 3} – {2, 3, 4} = {1}Remove elements of B from A

Prove: A – B ABy definition, A – B A xA–B xA

1. x A – B premise

2. x A x B definition

3. x A simplification

A B

Set Operations: Complement

Complement~ A E – A {x | xE xA}~{1, 2, 3} = {4} if E = {1, 2, 3, 4}

Prove: A ~A = A ~A =

A ~A A ~A set equality A ~A T is subset of every set A ~A identity x A x ~A x def of and x A x E x A x def of ~ F x comm., contradict.,

dominat. T

Note: Unary operators have precedence over binary operators. Use parentheses for the rest. Possible to define precedence: ~, , , .

A

Basic Set IdentitiesSet Algebra Name

A ~A = EA ~A =

Complementation lawExclusion law

A E = AA = A

Identity laws

A E = EA =

Domination laws

A A = AA A = A

Idempotent laws

Duals: and E

Basic Set Identities (continued…)Set Algebra Name

~(~A) = A Double Compl.

A B = B A A B = B A

Commutative laws

(A B) C = A (B C) (A B) C = A (B C)

Associative laws

A (B C) = (A B) (A C) A (B C) = (A B) (A C)

Distributive laws

~ (A B) = ~A ~B~ (A B) = ~A ~B

De Morgan’s laws

Note the similarity with logic

Example: Set Laws• Absorption

A (A B) = AA (A B) = A

• Venn Diagram “Proof”

Prove: A (A B) = A A (A B)

= (A ) (A B) ident.= A ( B) distrib.= A dominat.= A ident.

A B

Every set is a predicate

Every predicate is a set

Tuples• Things (usually a small number) arranged in order• 2-tuples

– Also known as pairs: (x, y)– Ordered: (x, y) (y, x) unless x = y

• n-tuples– n elements in order: (x1, x2, …, xn)

• Typically, elements are taken from known sets– x females, y males

(Mary, Jim) – might mean: Mary and Jim are a married couple

– x people, y cars(Mary, red sports car17) – might mean: Mary owns red sports car17

– x, y, z integers(3, 4, 7) – might mean: 3 + 4 = 7

Cartesian Product

• A B = {(x, y) | xA yB}

e.g., A = {1, 2}B = {a, b, c}A B = {(1, a), (1, b), (1, c),

(2, a), (2, b), (2, c)}

• |A B| = |A| · |B| = 2 · 3 = 6

Cartesian Product (cont’d)• n-fold Cartesian Product

A1 … An = {(x1, …, xn) | xA1 … xnAn}e.g., A = {1, 2}

B = {a, b, c}C = {, }

A B C = {(1,a,), (1,a,), (1,b,), (1,b,), (1,c,), (1,c,), (2,a,), (2,a,), (2,b,), (2,b,), (2,c,), (2,c,)}

• Can get large:A = set of students at BYU (30,000)B = set of BYU student addresses (10,000)C = set of BYU student phone#’s (60,000)|A| |B| |C| = 1.8 1013

Relations• Relation

– Subset of the cross product– Not necessarily a proper subset– R A B or R A B C

• Examples:– A = {1, 2} & B = {a, b, c}

R = {(1, a), (2, b), (2, c)}– A = {1, 2} & B = {a, b, c} & C = {, }

R = {(1, a, ), (2, c, )}– Marriage: subset of the cross product of

males and females

Relational Data Model

Relational Database• A persistent collection of relations

– Information about various kinds of objects (persons, places, things, events)

• Each relation holds information about various kinds of objects (persons, places, things, events, etc.)

• Each relation, or table, is characterized by a set of attributes or properties from a domain

e.g., person(SSN, Name, Address, Phone)

• Each individual object, or record, is a tuple of valuese.g., (123-45-6789, Pat Carter, 12 Main, 555-5555)

• Set of records make up the relation, i.e., subset of the cross-product of the attributes’ domains

Assume DB exists – in reality, create tables, insert tuples, etc.

DBMS – manage storage, integrity/security, crash recovery, etc.

StudentID Name Address Phone

12345 C. Brown 12 Apple St. 555-1234

67890 L. Van Pelt 34 Pear Ave. 555-5678

22222 P. Patty 56 Grape Blvd. 555-9999

33333 Snoopy 12 Apple St. 555-1234

Course StudentID Grade

CS101 12345 A

CS101 67890 B

EE200 12345 C

EE200 22222 B+

EE200 33333 B

CS101 33333 A-

PH100 67890 C+

Course Day Hour

CS101 M 9AM

CS101 W 9AM

CS101 F 9AM

EE200 Tu 10AM

EE200 W 1PM

EE200 Th 10AM

PH100 Tu 11AM

Course Prerequisite

CS101 CS100

EE200 EE005

EE200 CS100

CS120 CS101

CS121 CS120

CS205 CS101

CS206 CS121

CS206 CS205

Course Room

CS101 Turing Aud.

EE200 25 Ohm Hall

PH100 Newton Lab.

Relational Database Examplesnap cr

cdh csgcp

Relational Schemas• Each table has a schema

– Name– Set of attributes– Domain for each attribute

• Example: – Names: snap, cp, cdh, cr, csg– Attributes: table headers– Domains: studentID: integer

all the rest are strings, but we could be more specific (e.g. time, day, grade)

Relational Tables• Tables consist of n-tuples, where n is the arity or

degree of the relation (i.e., the number of attributes)

• Each n-tuple t D1 D2 … Dn, where the Di’s are the domainse.g., a 3-tuple t of cdh is an element string string string

or string day time or course day time depending on how specific we make our domains

• A table is a set of tuples, all with the same schemae.g., cdh Dcourse Dday Dhour

Tables & Keys

• Because a table is a set of tuples, there are no duplicates• There is always a set of attributes whose values uniquely

identify a tuple (even if it is all of them)• A set of attributes whose values always uniquely identify a

tuple constitutes a key• Typically, one or two attributes make up a key• Keys must be declared: we cannot assume uniqueness

– e.g., Name is not a key; there could be another C. Brown

• Some systems add a tuple identifier as the key

Keys ExamplesTable Key

• snap StudentIDName, Address, Phone (possible key?)

• cp Course Prerequisite

• cdh Course Day HourCan a course meet twice on the same day?If not: Course Day

• cr Course RoomDoes a course always meet in the same room?If so: Course

• csg Course StudentID

Predicates and Tuples• A table name for tuples of arity n is an n-place

predicate– cdh('CS101','M','9AM')– Asserts that CS101 meets on Monday at 9:00 am

• Predicates give each tuple a meaning in the ordinary sense of predicates– The subset of D1 D2 … Dn present in the database

are those assigned T; all others are assigned F (CWA)– Interpretation:

• Domain• For each predicate and every substitution, T or F

Every relation is a set, every set is a predicate, hence every relation is a predicate

And vice-versa

Database Tuples• Database tuples (strictly speaking) are not true

subsets of D1 D2 … Dn because we can alter the column order if we do so “correctly”

• More properly defined, a tuple in a relation is a set of attribute-value pairs– e.g. {(Course, 'CS101'), (Day, 'M'), (Hour, '9AM') }

= {(Day, 'M'), (Course, 'CS101'), (Hour, '9AM') }– Normally, we factor out the attribute and fix the order– Implication: we can interchange columns

cr = Course Room = Room Course CS101 Turing Aud. Turing Aud. CS101

EE200 25 Ohm Hall 25 Ohm Hall EE200 PH100 Newton Lab. Newton Lab. PH100