Lecture 5 Set Theory

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 2

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 3

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 4

Sets – indices Talk about arbitrary elements, where each subscript is a different integer: – {a i, a j,..., a n } Talk about systematically going through the set, where each superscript is a different integer: – {a 1 i, a 2 j,..., a 7 n } – {Edinburgh 1, Perth 2, Dundee 3, Aberdeen 4, Glasgow 5 } 5

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,...}? 6

Membership and definition of sets Membership of a set – a  S – represents that a is an element of set S – a  S – represents that a is 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,...} 7

Why set theory? Example: check if an element occurs in a collection 8 begin input x, a 1, a 2,...,a n ; if x  {a 1, a 2,...,a n } then found := true; end begin input x, a 1 4, a 2 2,...,a 8 7 ; found := false; for i := 1 to n do if x = a i j then found := true and output found; else output found; end search though collection by superscript.

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 9

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. 10

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)) 11

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 Venn diagrams 12 A B C D D  CD  C John Venn

Set equality (1) Two sets are equal if they have the same elements Formally, A and B are equal if A  B and B  A That is,  x ((x  A)  (x  B)) and  y ((y  B)  (y  A)) We represent this as A = B 13

Solution If x  A then x 2 is an odd integer, which means x is odd (this needs a proof, but let’s assume it has been proven). Therefore, x  B and so A  B. Solution If x  A then x 2 is an odd integer, which means x is odd (this needs a proof, but let’s assume it has been proven). Therefore, x  B and so A  B. Conversely, if x  B then x is an odd integer, and x 2 is an odd integer (this also needs a proof, but again let’s assume it has been proven). Therefore, x  A and so B  A. Set equality (2) Let A = {n : n 2 is an odd integer} Let B = {n : n is an odd integer} Show that A = B 14

Proof has two parts Part 1: all elements of A are elements of B Part 2: all elements of B are elements of A Set equality (3) 15

The union of sets A and B is A  B = {x : x  A or x  B} That is, – Those elements belonging to A together with – Those elements belonging to B and – (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} Set operations: union (1) 16

Venn diagram (area of interest in grey) Set operations: union (2) 17 A B A  BA  B

The intersection of sets A and B is A  B = {x : x  A and x  B} That is, – Only those elements belonging to both A and B Examples: {1, 2, 3, 4}  {4, 3, 2, 1} = {1, 2, 3, 4} {a, b, c}  {1, 2} = { } =  (empty set) Set operations: intersection (1) 18

Venn diagram (area of intersection in darker grey) Set operations: intersection (2) 19 A A  BA  B B

The complement of a set B relative to a set A is A – B = A \ B = {x : x  A and x  B} That is, – Those elements belonging to A and not 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} Set operations: complement (1) 20

Venn diagram (area of interest in darker grey) Set operations: complement (2) 21 A B A – B

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} Universal set 22 A

Symmetric difference of two sets A and B is A  B = {x : (x  A and x  B) or (x  B and x  A)} That is: – Elements in A and not in B or – Elements in B and not in A Or: elements in A or B, but not in both (grey area) Set operations: Symmetric difference 23 A B

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 Examples 24

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 Examples 25

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 Examples 26

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 Examples 27

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! Examples 28

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 = {name i,...., name n } – ID = {id i,...., id n } – COURSE = {course i,...., course n } – Student Info: (name j, id k, courses), where name j  NAME, id k  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}),...} Information modelling with sets 29

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)  R and CS1022  Courses} = {(bob,345), (mary,222), (mary,247),...} Query the Student Database 30

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 Summary 31

Further reading R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd (Chapter 3) Wikipedia’s entry Wikibooks entry 32