263-2200
Types and Programming
Languages
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Outline
Boolean Algebra
Values and Operations
Axioms and Theorems
Set Theory
Set Notation and Operations
Relations
Multisets and Sequences
Multisets
Sequences and Tuples
Graphs and Trees
Graphs
Trees
Directed Graphs
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Outline
Boolean Algebra
Values and Operations
Axioms and Theorems
Set Theory
Set Notation and Operations
Relations
Multisets and Sequences
Multisets
Sequences and Tuples
Graphs and Trees
Graphs
Trees
Directed Graphs
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Review of Boolean Algebra
I
Boolean Value: true = 1
I
Boolean Value: false = 0
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Boolean Type = {true, false} = {1, 0} = all Boolean values
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A proposition is a statement that is either true or false.
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Boolean Operations
conjunction
disjunction
negation
implication
implication
equivalence
and
or
not
only-if
if
if-and-only-if
∧
∨
¬
→
←
↔
∗
+
−
0
≡
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Signatures of Boolean Operators
and
or
not
→
←
↔
:
:
:
:
:
:
Boolean ∗ Boolean → Boolean
Boolean ∗ Boolean → Boolean
Boolean → Boolean
Boolean ∗ Boolean → Boolean
Boolean ∗ Boolean → Boolean
Boolean ∗ Boolean → Boolean
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Precedence of Boolean Operators
Level 1 (lowest):
Level 2:
Level 3:
Level 4:
Level 5 (highest):
↔
→, ←
∨
∧
¬
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Truth Table Semantics of Some Basic Operations
F
0
0
1
1
G
0
1
0
1
¬F
1
1
0
0
F ∧G
0
0
0
1
F ∨G
0
1
1
1
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Equational Semantics of Some Basic Operations
F ↔G
F ←G
F →G
=
=
=
(F → G) ∧ (F ← G)
G→F
¬F ∨ G
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Boolean Axioms
Commutative =
(p ∧ q) ∧ r = p ∧ (q ∧ r )
(p ∨ q) ∨ r = p ∨ (q ∨ r )
p ∧ (q ∨ r ) = (p ∧ q) ∨ (p ∧ r )
p ∨ (q ∧ r ) = (p ∨ q) ∧ (p ∨ r )
Associative =
Distributive =
p∧q =q∧p
p∨q =q∨p
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Identity =
p ∧ true = p
p ∨ false = p
Negation =
p ∨ ¬p = true
p ∧ ¬p = false
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Boolean Theorems and Observation
Double Negation =
Universal =
¬(¬p) = p
p∧p =p
p∨p =p
¬(p ∧ q) = ¬p ∨ ¬q
¬(p ∨ q) = ¬p ∧ ¬q
Idempotent =
De Morgan =
p ∨ true = true
p ∧ false = false
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p ∨ (p ∧ q) = p
p ∧ (p ∨ q) = p
¬true = false
¬false = true
Absorption =
Negation II =
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Outline
Boolean Algebra
Values and Operations
Axioms and Theorems
Set Theory
Set Notation and Operations
Relations
Multisets and Sequences
Multisets
Sequences and Tuples
Graphs and Trees
Graphs
Trees
Directed Graphs
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Basic Values and Notation
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Empty Set = {} = ∅
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Universal Set = U
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The set of all elements x satisfying the Boolean formula
(i.e., the predicate) P is denoted:
{x | P(x)}
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Other Notations for Describing Sets
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Dot-dot notation:
{1, 2, 3, ...}
{1, 2, 3, ..., 10}
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Regular Expressions
I
Context-free Grammars
Set type – all legal sets including the empty set and the
universal set.
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Set operations
member
intersection
union
compliment
subset
proper subset
superset
proper superset
equivalence
difference
power set
cardinality
∈
∩
∪
A
⊆
⊂
⊇
⊃
≡
−
P(A)
|A|
or Ac
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Signatures of set operators:
∈
∩
∪
A
⊆
⊂
⊇
⊃
≡
−
P(A)
|A|
:
:
:
:
:
:
:
:
:
:
:
:
element ∗ set → Boolean
set ∗ set → set
set ∗ set → set
set → set
set ∗ set → Boolean
set ∗ set → Boolean
set ∗ set → Boolean
set ∗ set → Boolean
set ∗ set → Boolean
set ∗ set → set
set → set
set → integer when A is a finite set
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Precedence of set operators:
Remark
Oftentimes parenthesis is the only convention used to indicate
order of evaluation.
Level 1 (lowest):
Level 2:
Level 3:
Level 4:
Level 5:
Level 6 (highest):
≡
⊆, ⊂, ⊇, ⊃
−
∪
∩
A
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Semantics of set operators:
A∪B
A∩B
A
=
=
=
{x|x ∈ A ∨ x ∈ B}
{x|x ∈ A ∧ x ∈ B}
{x|x 6∈ A} = U − A
A−B
=
=
{x|x ∈ A ∧ ¬(x ∈ B)} = {x|x ∈ A ∧ x 6∈ B}
A∩B
A≡B
A⊆B
A⊂B
A⊇B
A⊃B
P(A)
|A|
=
=
=
=
=
=
=
∀x : x ∈ A ↔ x ∈ B
∀x : x ∈ A → x ∈ B
(∀x : x ∈ A → x ∈ B) ∧ A 6≡ B
B⊆A
B⊂A
{s|s ⊆ A}
the number of elements in A
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Remarks
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Terminology: Two sets are disjoint if A ∩ B = ∅
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The tuple ( ∅, U, union, intersection, compliment ) forms a
Boolean algebra – meaning the boolean axioms and
theorems hold for sets.
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Review of Relations
A relation, R, on sets A and B is a set of ordered pairs.
R = {(a, b)|a ∈ A ∧ b ∈ B}
Oftentimes we write a R b to mean (a, b) ∈ R.
Definition
Let R be a binary relation on a set A.
1. R is reflexive iff ∀a ∈ A : a R a
2. R is symmetric iff ∀a, b ∈ A : a R b → b R a
3. R is antisymmetric iff ∀a, b ∈ A : a R b ∧ b R a → a = b
4. R is transitive iff ∀a, b, c ∈ A : a R b ∧ b R c → a R c
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Definition
Let R be a binary relation on a set A.
1. R is a preorder iff R is reflexive and transitive.
2. R is a partial order relation iff R is reflexive, antisymmetric,
and transitive.
3. R is an equivalence relation iff R is reflexive, symmetric,
and transitive.
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Outline
Boolean Algebra
Values and Operations
Axioms and Theorems
Set Theory
Set Notation and Operations
Relations
Multisets and Sequences
Multisets
Sequences and Tuples
Graphs and Trees
Graphs
Trees
Directed Graphs
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Multisets
Definition
A multiset is an extension to the notion of a set in which
collections can contain repeated elements. For example, {3, 8}
and {3, 8, 8} denote two different multisets. However, {3, 8, 8}
and {8, 8, 3} denote the same multiset.
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Sequence
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A sequence of objects is a list of these objects in some
order.
I
Enclosing a comma-separated list of objects in parenthesis
indicates that the list is a sequence.
I
A sequence can be seen as an extension to the notion of a
multiset where the notion of order (or position in the list)
matters.
I
Finite sequences are also called tuples.
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A sequence having k elements is a k -tuple.
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Outline
Boolean Algebra
Values and Operations
Axioms and Theorems
Set Theory
Set Notation and Operations
Relations
Multisets and Sequences
Multisets
Sequences and Tuples
Graphs and Trees
Graphs
Trees
Directed Graphs
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An undirected graph, or simply a graph, can be defined by
G = (V , E) where V : Id and E : Id ∗ Id.
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Elements of V are called nodes or vertices.
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Elements of E are called edges.
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An edge is a tuple of the form V ∗ V .
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In an undirected graph, the order of the elements in an
edge tuple does not matter. That is, (vi , vj ) = (vj , vi ).
Another way of thinking about this is that the pair of
vertices in an edge is represented as a set (and not a
sequence/tuple).
I
A graph can be pictorially represented by drawing elements
of V as labelled dots (or small circles) and elements of E
as lines connecting corresponding labelled dots.
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I
The degree of a node v ∈ V is equal to the number of
edges in e ∈ E in which v occurs. Pictorially, it is equal to
the number of lines that connect v to other nodes in the
graph.
I
A subgraph of a graph G1 = (V1 , E1 ) is a graph
G2 = (V2 , E2 ) such that V2 ⊆ V1 ∧ E2 ⊆ E1 .
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A path in a graph is a sequence of nodes connected by
edges. Formally, given G = (V , E) the sequence
(v1 , v2 , ..., vn ) is a path if
vi ∈ V ∧ ∀i : 1 ≤ i < n → (vi , vi+1 ) ∈ E ∨ (vi+1 , vi ) ∈ E.
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A path (v1 , v2 , ..., vn ) is a simple path if vi = vj ↔ i = j.
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I
A graph G = (V , E) is connected if it is true that ∀vi , vj ∈ V
there exists a path of the form (vi , ..., vj ).
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A path (v1 , v2 , ..., vn ) is a cycle if v1 = vn .
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A simple cycle is a cycle (v1 , v2 , ..., vn ) such that n ≥ 3
which contains no subsequences that are themselves
cycles.
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Trees
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A tree is a graph that is connected and has no simple
cycles.
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In a tree, nodes of degree 1 are called the leaves of the
tree.
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In a tree, a special node is (often) designated to be the
root.
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Directed Graphs
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A directed graph is a graph in which the order of vertices in
edges matters. That is, (vi , vj ) 6= (vj , vi ).
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A directed edge (vi , vj ) can be pictorially represented by
drawing an arrow from vi to vj .
I
Given a directed graph G = (V , E). The outdegree of a
node v1 ∈ V is equal to the number of edges e ∈ E of the
form (v1 , v2 ). The indegree is equal to the number of
edges of the form (v2 , v1 ).
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Directed Graphs
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Given a directed graph G = (V , E). A path (v1 , v2 , ..., vn ) is
said to be a directed path if ∀i : 1 ≤ i < n → (vi , vi+1 ) ∈ E
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A directed graph is strongly connected if a directed path
connects every two nodes.
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