Download Discrete Mathematics - Department of Computer Sciences

1
Conversion
Algorithms
Binary
& Hex
Induction
Recurrence
Equations
Signed
Integers
Summations
Recursion &
Induction
Numbers
& Naming
Floating
Point
Sequences
Discrete
Mathematics
Equivalences
Boolean
(Propositional)
Partial
Orders
Relations &
Functions
Logic
& Sets
First-Order
(Predicate)
Polynomials
Sets &
Subsets
Boolean
& Set
Operations
Permutations
Logs &
Exponentials
Integer
Functions
Figure 1: Mind map for discrete mathematics
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Direct
Mersenne
Fibonacci
Inference
Rules
Indirect
Two Dimensional
Contradiction
Triangular
Sequences
Proofs
Geometric
Induction
Arithmetic
Discrete
Mathematics
Truth
Assignments
Euclidean
Algorithm
Boolean
Expressions
Number
Theory
Counting
Congruence
Equations
Relations
Subsets
Functions
Prime
Numbers
Modular
Numbers
Figure 2: Mind map for discrete mathematics
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OR: ∨
NOT: ¬
AND: ∧
IMPLIES:
→
Predicate
Laws
Boolean
Intersect: ∩
There
exists: ∃
Union: ∪
Complement: ¬
Set
Predicates
Operations
Cartesian
Product: ×
For all: ∀
Logic & Sets
Diagrams
True or
False
Cardinality
Propositions
Sets &
Subsets
Subset
Symbol: ⊆
Variables:
p, q, r
Boolean
Laws
Partitions
Pascal’s
Triangle
Truth Tables
Second
Stirling
Numbers: {n
k}
Factorials
Binomial
Coefficients
Figure 3: Mind map for logic and sets
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Function:
m(n) =
2n − 1
h0, 1, 3, 7i
Binomial Coefficients: (nk)
Recurrence:
mn =
m n −1 + m n −2
Tower of
Hanoi
h0, 1, 1, 2i
Mersenne
Function:
f (n) = ( ϕn −
√
ϕ̄n )/ 5
First Stirling
Numbers: [nk]
Second
Stirling
Number: {nk}
Two Dimensional
Fibonacci
Recurrence:
fn =
f n −1 + f n −2
Sequences
Recurrence:
an =
a n −1 + m
h0, 0, 1, 3i
Arithmetic
Triangular
Function:
t(n) =
n(n − 1)/2
Function:
a(n) =
mn + b
Geometric
Recurrence:
tn =
t n −1 + ( n − 1 )
hb, m + b, 2m + b, . . .i
Recurrence:
gn = rgn−1
a, ar, ar2 , . . .
Function:
g(n) = ar n
Figure 4: Mind map for sequences
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Recurrence
Initial
Conditions
Functions
Solving
Recurrences
Recurrence
Equations
Recursion
Recursion &
Induction
Induction
Basis
Conclusion
Hypothesis
Figure 5: Mind map for recursion and
induction
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Domain Cardinality: |X|
Co-Domain
Cardinality: |Y|
Binomial Coefficients: (nk)
Cardinality
of Set: n
2|X||Y|
Relations
Subsets
Co-Domain
Cardinality: |Y|
Cardinality
of Subset: k
Counting
2n truth assignments on
n variables
Count of
Boolean
Variables: n
Functions
Domain Cardinality: |X|
| Y | |X|
Truth Assignments
Boolean
Expressions
(Functions)
n
22 Boolean
functions on
n variables
Count of
Boolean
Variables: n
Figure 6: Mind map for counting
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Subset of
ordered pairs
Subset ⊆
Adjacency
matrix
Bipartite
graph
Divides |
Partial
Orders
Representing
Less than
or equal ≤
Relations
Reflexive:
x ∼ x
Equivalences
Antisymmetric:
x ∼ y∧y ∼ x →
x=y
Properties
Congruence
mod n
Parallel
Projections
Transitive:
x ∼ y∧y ∼ z →
x∼z
Nondeterministic
Symmetric:
x∼y →
y∼x
Figure 7: Mind map for relations
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Cyclic
Notation
First Stirling
Numbers: [nk]
n! of n
Things
Ceiling d x e
Floor b x c
Permutations
GCD
Integer
Functions
Mod
(Remainder)
(logb x =
loga x
loga b
Quotient
Logs & Exponentials
Functions
(lg x = y) ≡
(2y = x )
Polynomials
Deterministic
Properties
Horner’s
Scheme
Domain,
Co-Domain,
Range
Inverse
Onto
One-to-one
Figure 8: Mind map for functions
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Sign,
Exponent,
Fraction
Normalization
Scientific
Notation
Horner’s
Scheme
Floating
Point
Remaindering
Conversion
Algorithms
Biased
Notation
Numbers
& Naming
Signed
Integers
Complement
Notation
Alphabet
Languages
Unsigned
Integers
String
Hex:
{0, 1, . . . , E, F }
Word
Binary:
{0, 1}
Decimal:
{0, 1, . . . , 9}
Figure 9: Mind map for numbers and
naming
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Extended
Euclidean
Algorithm
Multiplicative Inverse
(Reciprocal)
Relatively
Prime
Euclidean
Algorithm
Linear
Congruence
Equations
Greatest
Common
Divisor
Numbers Theory
Fundamental
Theorem of
Arithmetic
Prime
Numbers
Modular
Numbers
Multiplication
Unbounded
Set
Composite
Numbers
Addition
Figure 10: Mind map for number
theory