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### What is a Graph? how highly connected is an entity within a network? What is an entity's overall importance in a network ?

INSTRUCTIONS TO CANDIDATES

## What is a Graph?

• A graph is a data structure representing connections between items
• We're storing values with the potential for connections between any or all elements
• Examples of graphs in everyday life:
• PERT charts
• Flow charts
• Examples in computer science
• Networks
• Social Network diagrams
• Executing a makefile

Social Networks

Using Social Network Analysis, you can answer:

• How highly connected

is an entity within a network?

• What is an entity's

overall importance in a network?

• How central is an entity within a network?
• How does information

flow within a network?

Makefiles

How make Works

• Construct the dependency graph from the target and dependency entries in

the makefile

• Do a topological sort to determine an order in which to construct
• For each target visited, invoke the commands if the target file does not exist or if any dependency file is newer
• Relies on file modification

dates

## Set Theory

• A set is any collection of objects, e.g. a set of vertices
• The objects in a set are called the elements of the set

• Repetition and order are not important
• {2, 3, 5} = {5, 2, 3} = {5, 2, 3, 2, 2, 3}

• Sets can be written in predicate form:
• {1, 2, 3, 4} = {x : x is a positive integer less than 5}
• The empty set is {} = 0, all empty sets are equal

## Elements and Subsets

• x ∈ A means “x is an element of A”
• x ∈ A means “x is not an element of A”

• A ⊆ B means “A is a subset of B”
• x ⊆ A means “x is not a subset of A”

• If A ⊆ B and A ≠ B…

A is a proper subset of B and we write A ⊂                                                                               B

• A ⊆ A for every set A
• Every set is a subset of itself

## Subsets

• If A ⊆ B and B ⊆  C then A ⊆  C
• If A ⊆ B and B ⊆ A then A = B

• The empty set is a subset of every set:

⊆ A for any A

• The subsets of A ={1, 2, 3} are:

,{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}

(Sometimes called the powerset of A, P(A) )

## Operations on Sets

• Union
• A ∪ B = { x : x ∈ A or x ∈ B }

• Intersection
• A ∩ B = { x : x ∈ A and x ∈ B }

• Universal Set
• All sets under consideration will be subsets of a background set, called the Universal Set, U
• Complement
• A' = { x : x ∈ U and x ∈ A }

## Example

• Let:
• U = {a, b, c, d, e, f}
• A = {a, c}
• B = {b, c, f}
• C = {b, d, e, f}

• Then:
• B ∪ C = {b, c, d, e, f}
• A ∩ (B ∪ C) = {c}
• A' = {b, d, e, f}

= C

• A' ∩ (B ∪ C) = C ∩ (B ∪  C) = {b, d, e, f} = C
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