1. CS1022 Computer Programming & Principles
Lecture 1
Set Theory

2. 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
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3. 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
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4. 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}
{CS1024, 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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5. 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}
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6. 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, ...}?
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7. 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, ...}
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8. Why set theory? Example: check if an element occurs in a collection
begin
input x, a14, a22, ...,a87;
found := false;
for i := 1 to n do
if x = aij then found := true
and output found;
else output found;
end
search though
collection by
superscript.
begin
input x, a1, a2, ...,an;
if x  {a1, a2, ...,an} then found := true;
end
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9. 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
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10. 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.
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11. A  B if, and only if, x ((x  A) (x  B))
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))
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12. 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 A 1 2 3
472 B
John Venn C D
D  C
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13. x ((x  A) (x  B)) and y ((y  B) (y  A))
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
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14. Set equality (2) Let A = {n : n2 is an odd integer}
Let B = {n : n is an odd integer}
Show that A = B
Solution
If x  A then x2 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 x2 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.
Solution
Solution
If x  A then x2 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.
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15. 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
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16. Set operations: union (1)
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}
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17. Set operations: union (2)
Venn diagram (area of interest in grey) A B
A  B
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18. Set operations: intersection (1)
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)
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19. Set operations: intersection (2)
Venn diagram (area of intersection in darker grey) A B
A  B
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20. Set operations: complement (1)
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}
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21. Set operations: complement (2)
Venn diagram (area of interest in darker grey) A B
A – B
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22. 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
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23. Set operations: Symmetric difference
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) A B
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24. 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
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25. 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
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26. 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
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27. 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
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28. 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!
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29. 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}), ...}
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30. 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)  R and CS1022  Courses}
= {(bob,345), (mary,222), (mary,247), ...}
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31. 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
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32. Further reading
R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd (Chapter 3)
Wikipedia’s entry
Wikibooks entry
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