CS1022 Computer Programming & Principles

CS1022Computer Programming & Principles Lecture 3.1 Set Theory (1)

Plan of lecture • Why set theory? • Sets and their properties • Membership and definition of sets • “Famous” sets • Types of variables and sets • Sets and subsets • Venn diagrams • Set operations CS1022

Why set theory? • Set theory is a cornerstone of mathematics • Provides a convenient notation for many concepts in computing such as lists, arrays, etc. and how to process these CS1022

Sets • A set is • A collection of objects • Separated by a comma • Enclosed in {...} (curly brackets) • Examples: • {Edinburgh, Perth, Dundee, Aberdeen, Glasgow} • {2, 3, 11, 7, 0} • {CS1015, CS1022, CS1019, SX1009} • Each object in a set is called an element of the set • We use italic capital letters to refer to sets: • C = {2, 3, 11} is the set C containing elements 2, 3 and 11 CS1022

Sets – indices • Talk about arbitrary elements, where each subscript is a different integer: • {ai, aj, ..., an} • Talk about systematically going through the set, where each superscript is a different integer: • {a1i, a2j, ..., a7n} • {Edinburgh1, Perth2, Dundee3, Aberdeen4, Glasgow5} CS1022

Properties of sets • The order of elements is irrelevant • {1, 2, 3} = {3, 2, 1} = {1, 3, 2} = {2, 3, 1} • There are no repeated elements • {1, 2, 2, 1, 3, 3} = {1, 2, 3} • Sets may have an infinite number of elements • {1, 2, 3, 4, ...} (the “...” means it goes on and on...) • What about {0, 4, 3, 2, ...}? CS1022

Membership and definition of sets • Membership of a set • a S– represents that ais an element of set S • a  S – represents that ais not an element of set S • For large sets we can use a property (a predicate!) to define its members: • S = {x : P(x)} – S contains those values for x which satisfy property P • N = { x : x is an odd positive integer} = {1, 3, 5, ...} CS1022

Why set theory? • Example: check if an element occurs in a collection search though collection by superscript. CS1022

Names of “famous” sets • Some sets have a special name and symbol: • Empty set: has no element, represented as { } or  • Natural numbers: N = {1, 2, 3, ...} • Integers: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...} • Rational numbers: Q = {p/q : p, q  Z, q  0} • Real numbers: R = {all decimals} • N.B.: in some texts/books 0 N CS1022

Types (of variables) and sets • Many modern programming languages require that variables be declared as belonging to a data type • A data type is a set with a selection of operations on that set • Example: type “int” in Java has operations +, *, div, etc. • When we declare the type of a variable we state what set the value of the variable belongs to and the operations that can be applied to it. CS1022

Sets and subsets • Some sets are contained in other sets • {1, 2, 3} is contained in {1, 2, 3, 4, 5} • N (natural numbers) is contained in Z (integers) • Set A is a subset of set B if every element of A is in B • We represent this as A  B • Formally, A  B if, and only if, x ((x  A) (x  B)) CS1022

Venn diagrams • A diagram to represent sets and how they relate • A set is represented as an oval, a circle or rectangle • With or without elements in them • Venn diagrams show area of interest in grey • Venn diagram showing a set and a subset D C A B 3 1 472 2 John Venn D C CS1022

Set equality (1) • Two sets are equal if they have the same elements • Formally, A and B are equal if A  Band B  A • That is, x ((x  A) (x  B)) and y ((y  B) (y  A)) • We represent this as A = B CS1022

Set equality (2) • Let A = {n : n2 is an odd integer} • Let B = {n : n is an odd integer} • Show that A = B CS1022

Set equality (3) Proof has two parts • Part 1: all elements of A are elements of B • Part 2: all elements of B are elements of A CS1022

Set operations: union (1) • The union of sets A and B is A  B = {x : x  Aorx  B} • That is, • Those elements belonging to Atogether with • Those elements belonging to Band • (Possibly) those elements belonging to both A and B • N.B.: no repeated elements in sets!! • Examples: {1, 2, 3, 4}  {4, 3, 2, 1} = {1, 2, 3, 4} {a, b, c}  {1, 2} = {a, 1, b, 2, c} CS1022

Set operations: union (2) • Venn diagram (area of interest in grey) B A A B CS1022

Set operations: intersection (1) • The intersection of sets A and B is A  B = {x : x  Aandx  B} • That is, • Only those elements belonging to both AandB • Examples: {1, 2, 3, 4}  {4, 3, 2, 1} = {1, 2, 3, 4} {a, b, c}  {1, 2} = { } =  (empty set) CS1022

Set operations: intersection (2) • Venn diagram (area of intersection in darker grey) A B A B CS1022

Set operations: complement (1) • The complement of a set B relative to a set A is A – B = A \ B = {x : x  Aandx  B} • That is, • Those elements belonging to Aandnot belonging to B • Examples: {1, 2, 3, 4} – {4, 3, 2, 1} = { } =  (empty set) {a, b, c} – {1, 2} = {a, b, c} {1, 2, 3} – {1, 2} = {3} CS1022

Set operations: complement (2) • Venn diagram (area of interest in darker grey) B A A– B CS1022

Universal set • Sometimes we deal with subsets of a large set U • U is the universal set for a problem • In our previous Venn diagrams, the outer rectangle is the universal set • Suppose A is a subset of the universal set U • Its complement relative to U is U – A • We represent as U – A = A = {x : x  A} A CS1022

Set operations: Symmetric difference • Symmetric difference of two sets A and B is AB = {x : (x  Aandx  B)or(x  Bandx  A)} That is: • Elements in A and not in Bor • Elements in B and not in A Or: elements in A or B, but not in both (grey area) A B CS1022

Examples Let A = {1, 3, 5, 7} B = {2, 4, 6, 8} C = {1, 2, 3, 4, 5} Find • A  C • B  C • A – C • B C CS1022

Examples Let A = {1, 3, 5, 7} B = {2, 4, 6, 8} C = {1, 2, 3, 4, 5} Find • A  C = {1, 3, 5, 7}  {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5, 7} • B  C • A – C • B C CS1022

Examples Let A = {1, 3, 5, 7} B = {2, 4, 6, 8} C = {1, 2, 3, 4, 5} Find • A  C = {1, 3, 5, 7}  {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5, 7} • B  C = {2, 4, 6, 8}  {1, 2, 3, 4, 5} = {2, 4} • A – C • B C CS1022

Examples Let A = {1, 3, 5, 7} B = {2, 4, 6, 8} C = {1, 2, 3, 4, 5} Find • A  C = {1, 3, 5, 7}  {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5, 7} • B  C = {2, 4, 6, 8}  {1, 2, 3, 4, 5} = {2, 4} • A – C = {1, 3, 5, 7} – {1, 2, 3, 4, 5} = {7} • B C CS1022

Examples Let A = {1, 3, 5, 7} B = {2, 4, 6, 8} C = {1, 2, 3, 4, 5} Find • A  C = {1, 3, 5, 7}  {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5, 7} • B  C = {2, 4, 6, 8}  {1, 2, 3, 4, 5} = {2, 4} • A – C = {1, 3, 5, 7} – {1, 2, 3, 4, 5} = {7} • B C = (B – C)  (C – B) = ({2, 4, 6, 8} – {1, 2, 3, 4, 5})  ({1, 2, 3, 4, 5} – {2, 4, 6, 8}) = {6, 8}  {1, 3, 5} = {1, 3, 5, 6, 8} • N.B.: ordering for better visualisation! CS1022

Information modelling with sets • We can build an information model with sets • “Model” means we don’t care how it is implemented • Essence: what information is needed • Example: information model for student record • NAME = {namei, ...., namen} • ID = {idi, ...., idn} • COURSE= {coursei, ...., coursen} • Student Info: (namej, idk, courses), where namej NAME, idk ID,and courses  COURSE. • Student Database is a set of student info: R = {(bob,345,{CS1022,CS1015}), (mary,222,{SX1009,CS1022,MA1004}), (jill,246,{SX1009,CS2013,MA1004}), (mary,247,{SX1009,CS1022,MA1004}), ...} CS1022

Query the Student Database • R = {(bob,345,{CS1022,CS1015}), (mary,222,{SX1009,CS1022,MA1004}), (jill,246,{SX1009,CS2013,MA1004}), (mary,247,{SX1009,CS1022,MA1004}), ...} • Query to obtain a class list. Give set C, where: C= {(N,I) : (N,I,Courses)  Rand CS1022  Courses} = {(bob,345), (mary,222), (mary,247), ...} CS1022

Summary You should now know: • What sets are and how to represent them • Venn diagrams • Operations with sets • How to build information models with sets and how to operate with this model CS1022

Further reading • R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd. 2002. (Chapter 3) • Wikipedia’s entry • Wikibooks entry CS1022

CS1022 Computer Programming & Principles