Sets, Functions, Graphs, Trees

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Sets, Functions, Graphs, Trees
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Sets
A set is a collection of objects, called members
or elements
xÏS
xÎS
We can describe set by explicitly listing its
members
Sa,s,d,f,g
A set can not contain same object more than once
Two sets are equal if they contain same elements
A¹B
AB
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Some frequently encountered sets
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Subsets
If all the elements of a set A are contained in a
set B then A is a subset B
A Í B
A Í B
If A Í B and A ¹ B than A is proper subset B
A Ì B
AB if and only if A Í B and B Í A
For any A, B, and C if A Í B and B Í C , then A
Í C
For any A Æ Í A
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Definition of set in terms of other sets
Xx x/2 Î Z
Directly
Applying set operations
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Set operations laws
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Set operations laws II
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Set operations laws III
Venn diagram
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Universe
U
A
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DeMorgans laws
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Collection of sets
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Size (cardinality) of the set

• finite
• infinite
• countably infinite
• uncountably infinite

For any two finite sets A and B AÈB A
B - AÇB AÈB A B
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Cartesian product
A x B (a,b) a Î A, b Î B
A x B A B
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Binary relation
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Functions
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Functions
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Graphs
directed undirected
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Directed Graphs
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Undirected Graphs
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Adjacency relation
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Degree of a vertex
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Path
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Cycles
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Isomorphism
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Representation of graphs in memory (undirected
graph)
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Representation of graphs in memory (directed
graph)
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Tree Forrest
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Rooted Tree
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Height of the Tree
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Ordered Tree
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Ordered Tree
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Binary Tree
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Binary Tree II
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Binary Tree representation
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Binary Tree representation II