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- 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

- 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}

- {1, 2, 3, 4} = {x
- The empty set is {} = 0, all empty sets are equal

- 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

- 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) )

- 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 }

- 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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