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SET THEORY
Outline
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
What is a set?
Set Theory
– Union
– Intersection
– etc
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Sets in C
– Pseudocode
– Data Structures
– Code
What is a set?
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Set is an unordered collection of objects
Written as: {9,5,3,7,1}, {red, blue, white}
Can be
– finite number of elements (no. students in this
class)
– infinite (number of real numbers)
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Can contain any type of object
Order of listing of set elements not important
Duplicate elements do not appear
Examples
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{ 5, 6, 2, 1, 9, 8 }
okay?
– YES – unordered set of elements
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{ 1, 2, 3, 5, 8, 9 }
okay?
– YES – set of elements (order not important)
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{ 3, 7, 7, 9, 1, 5 }
okay?
– NO (Almost!) – wouldn’t contain duplicate
elements, e.g. two ‘7’s
John Venn (1857 – 1923)
Representing sets graphically
Venn diagrams:
U
A
1 5
9
3
7
Venn Diagram
Set:
A = { 5, 1, 7, 3, 9 }
Set terminology
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Empty set (no elements): 
– e.g. Set A = { }, A = 
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If an object x is in a set A, write x A
– e.g. Set A = { 1, 3, 5, 7, 9 }
x=7
Then x  A
(Read: “x is in the set A”)
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If not, write x  A
Examples
Set B = { 4, 9, 1, 2, 7, 6 }
X=8
Y=9
X B ?
X B ?
Y B ?
Y B ?
Universal sets
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Can define a universal set for particular area
Universal set contains all possible elements
under consideration
E.g. could define U = set of all positive ints
less than 10: U = {1,2,3,4,5,6,7,8,9}
U
A 1
59
37
Venn Diagram
Universal
set
Cardinality
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Number of elements a set contains
A = { 1, 2, 5, 6 }
Cardinality of A = 4
B = { 2, 17 }
Cardinality of B = 2
Set operations
Set intersection
Intersection of sets A and B, written A B, is a
new set containing only members common
to A and B
A = {1,3,5,7,9}
B = {1,2,3,4,5}
A B = ? {1,3,5}
Example
A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
B = { 21, 22, 10, 34, 12 }
A  B = ? { 10 }
Set operations : Union
Union of two sets A and B, written A B,
is a new set containing all members of A
and B
A = {1,3,5,7,9}
A B =
B = {1,2,3,4,5}
? {1,2,3,4,5,7,9}
Note: Elements not duplicated in a union
Example
A = { 1, 2, 3, 4 }
B = { 2, 1, 3, 5 }
C = { 3, 2, 1, 6 }
A  B  C = ? { 1, 2, 3, 4, 5, 6 }
Set difference
Difference of sets A and B, written A - B, is a
new set containing members of A not in B
A = {1,3,5,7,9}
B = {1,2,3,4,5}
A - B = ? {7,9}
Set difference
Difference of sets A and B, written A - B, is a
new set containing members of A not in B
A = {1,3,5,7,9}
B = {1,2,3,4,5}
B - A = ? {2, 4}
Subsets
Set A is a subset of another set B, if all
elements of A are contained within B.
E.g. B = {1,2,3,4,5,6,7,8,9}
A = {1,3,5,7}
then, A is a subset of B.
Notation: A  B
Venn Diagrams revisited
Venn diagrams
Represent a set by a circle:
U
A
1 5
9
3
7
A = { 1, 5, 9, 7, 3 }
Venn diagrams
U
A
B
10
1
5 9
11
3
13
7
12
B
10
9
11
13
12
?
A = { 1, 5, 3, 9, 7 }
B = { 10, 13, 9, 11, 12 }
Venn diagrams and operations
Set union A  B
U
B
A
10
1 5
9
11
3
13
7
12
?
A  B = { 1, 3, 5, 9, 7, 10, 11, 12, 13}
Venn diagrams
Set intersection A  B
U
B
A
10
1 5
9
11
3
13
7
12
?
AB={9}
Venn diagrams
Set difference A – B
U
B
A
10
1 5
9
11
3
13
7
12
?
A – B = { 1, 5, 3, 7 }
Venn diagrams
Subsets B  A
U
A
1 5
3
7
9
B
?
A = { 1, 5, 3, 7, 9}; B = { 3, 9 }

How would we represent sets using ‘c’ using
linked lists?
Summary
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Sets
– Unordered collection of objects
• No duplicates
– Terminology
• Empty set 
• x  A (“x is in the set A”)
– Cardinality
• Number of elements in a set
– Operations
• Union , intersection , difference -, subset 
– Representation : Venn Diagrams