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# CS1022 Computer Programming & Principles - PowerPoint PPT Presentation

CS1022 Computer 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. Why set theory?.

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### CS1022Computer Programming & Principles

Lecture 3.1

Set Theory (1)

• 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

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• 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

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• 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

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• 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}

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• 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, ...}?

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• 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, ...}

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• Example: check if an element occurs in a collection

search though

collection by

superscript.

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• 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

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• 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.

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• 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))

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• 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

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• 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

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• Let A = {n : n2 is an odd integer}

• Let B = {n : n is an odd integer}

• Show that A = B

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Proof has two parts

• Part 1: all elements of A are elements of B

• Part 2: all elements of B are elements of A

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• 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}

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• Venn diagram (area of interest in grey)

B

A

A B

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• 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)

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• Venn diagram (area of intersection in darker grey)

A

B

A B

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• 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}

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• Venn diagram (area of interest in darker grey)

B

A

A– B

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• 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

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• 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

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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

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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

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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

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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

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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!

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• 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}), ...}

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• 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), ...}

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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

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• R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd. 2002. (Chapter 3)

• Wikipedia’s entry

• Wikibooks entry

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