$\color{dodgerblue}\text{Discrete Definitions}$ $\textit{Author not responsible for consequences caused by any innacuracies.}$
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$\color{steelblue}\text{Equivalence}$ $$
\begin{align}
\text{Equals: } &\quad =\[1em]
\text{Greater than: } &\quad >\[1em]
\text{Greater than or equal to: } &\quad \ge\[1em]
\text{Less than: } &\quad <\[1em]
\text{Less than or equal to: } &\quad \le\[1em]
\text{Not equal to: } &\quad \not =\[1em]
\text{Modular equivalence: } &\quad \equiv\[1em]
\end{align}
$$
$\color{steelblue}\text{Operators}$ $$
\begin{align}
\text{Multiply: } &\quad \cdot\[1em]
\text{Mod: } &\quad \bmod\[1em]
\text{Implication: } &\quad \models\[1em]
\textit{NOT: } &\quad \lnot\[1em]
\textit{AND: } &\quad \wedge\[1em]
\textit{OR: } &\quad \vee\[1em]
\textit{XOR: } &\quad \oplus\[1em]
\textit{IF..THEN: } &\quad \rarr\[1em]
\textit{IF AND ONLY IF: } &\quad \harr\[1em]
\end{align}
$$
$\color{steelblue}\text{Formula Classification}$ $$
\begin{align}
\textbf{Tautologies: } &\quad always T\[1em]
\textbf{Contradictions: } &\quad always F\[1em]
\textbf{Contingent formulas: } &\quad T\ or\ F\quad \text{depending on variables} \[1em]
\textbf{Satisfiable formulas: } &\quad \textit{tautologies }\ or\ \textit{ contingent formulas}\[1em]
\end{align}
$$
$\color{steelblue}\text{Set Theory Abstractions}$ $$
\begin{align}
\text{Set: }&\quad S\[1em]
\text{Universe: }&\quad U\[1em]
\text{Member in set: }&\quad \in\[1em]
\text{Not member in set: }&\quad \notin\[1em]
\textbf{Real, } \textit{any number: }&\quad \Bbb R\[1em]
\textbf{Integer, } \textit{only whole: }&\quad \Bbb Z\[1em]
\textbf{Natural}\ _0\textbf{, } \textit{whole non-negatives: }&\quad \Bbb N_0\[1em]
\textbf{Natural}\ _1\textbf{, } \textit{whole positives: }&\quad \Bbb N_1\[1em]
\text{Subset: } &\quad A\subseteq B\[1em]
\text{Proper subset: } &\quad A\subset B\[1em]
\text{Equality through subsets: } &\quad S\subseteq T \And T\subseteq S\[2em]
\end{align}
$$
$\color{steelblue}\text{Set Constructors}$ $$
\begin{align}
\nonumber\textbf{Set listing: }\\
SMALLPRIMES=\ &\set{1,2,3,5,7}\[2em]
\nonumber\textbf{Implied pattern: }&\textit{(bad practice)}\\
EVENS=\ &\set{2,4,6,8,\dots}\[2em]
\nonumber\textbf{Setbuilder notation: }&\text{\textquotedblleft set comprehension"}\\
\nonumber\text{Subset of elements, }&\text{that match a condition}\\
\set{x\in S&:\phi(x)}\\
SQUARES=\ \set{x\in \Bbb{Z}&:x=y^2 \text{ for some }y\in \Bbb{Z}}\[2em]
\nonumber\textbf{Setbuilder notation: }&\text{\textquotedblleft replacement"}\\
\nonumber\text{Apply a theory to }&\text{each member and collect results}\\
\set{f(x)&:x\in S}\\
SQUARES=\ \set{x^2&:x\in \Bbb{Z}}\[2em]
\end{align}
$$
$\color{steelblue}\text{Set Operators}$ $$
\begin{align}
\nonumber\textbf{Union: }&\text{everything in either}\\
\nonumber\text{Must define }A\cup B &\text{ to be the smallest set, such that }A\subseteq S \And B\subseteq S\\
\text
A\cup B=\ &\set{x:x\in A\vee x\in B}\[1em]
\nonumber\textbf{Intersection: }&\text{everything in \textit{both} sides}\\
A\cap B=\ &\set{x\in A: x\in B}\[1em]
\nonumber\textbf{Difference: }&\text{remove items in one set from the other}\\
A\setminus B=\ &\set{x\in A: x\notin B}\[1em]
\end{align}
$$
$\color{steelblue}\text{Predicates}$ $$
\begin{align}
\text{Existential quantification, }\textit{there exists: }\quad \exists\[1em]
\text{Universal quantification, }\textit{for all:}\quad \forall\[1em]
\end{align}
$$
$\color{steelblue}\text{Exponents}$ $$
\begin{align}
\text{Exponent in }a^3: &\quad 3\[1em]
\text{Base in }a^3: &\quad a\[1em]
\text{kilo: }&\quad 2^{10}\[1em]
\text{mega: }&\quad 2^{20} = (2^{10})^2\[1em]
\text{giga: }&\quad 2^{30} = (2^{10})^3\[1em]
\text{tera: }&\quad 2^{40} = (2^{10})^4\[1em]
\text{peta: }&\quad 2^{50} = (2^{10})^5\[1em]
\text{exa: }&\quad 2^{60} = (2^{10})^6\[1em]
\end{align}
$$
$\color{steelblue}\text{Laws of Exponents}$ $$
\begin{align}
(ab)^n &= a^n \cdot b^n \[1em]
a^m \cdot a^n &= a^{m+n} \[1em]
a^{m-n} &= \dfrac{a^m}{a^n}\quad (\text{when } a \not&= 0) \[1em]
a^{-n} &= \dfrac{1}{a^n}\quad (\text{when } a \not&= 0) \[1em]
a^0 &= 1 \[1em]
(a^m)^n &= a^{m \cdot n} \[1em]
\end{align}
$$
$\color{steelblue}\text{Laws of Logarithms}$ $$
\begin{align}
\log_a 1 &= 0 \[2em]
\log_a a &= 1 \[2em]
\log_a (x \cdot y) &= \log_a x + \log_a y \[2em]
\log_a x^y &= y \log_a x \[2em]
\log_a \dfrac{1}{y} &= - \log_a y \[2em]
\log_a \dfrac{x}{y} &= \log_a x - \log_a y \[2em]
\log_b x &= (\log_b a) \cdot \log_a x \[2em]
\end{align}
$$
$\color{steelblue}\text{Ceiling and Floor}$ $$
\begin{align}
\text{Ceiling, round up:} &\quad \lceil a \rceil\[1em]
\text{Floor, round down:} &\quad \lfloor a \rfloor\[1em]
\end{align}
$$
$\color{steelblue}\text{Bits}$ $$
\begin{align}
\text{Bit string: } &\quad \overline{x}\[1em]
\text{Mod: } &\quad \bmod\[1em]
\texttt{NOT: } &\quad \backsim\[1em]
\texttt{AND: } &\quad \And\[1em]
\texttt{OR: } &\quad \mid\[1em]
\texttt{XOR: } &\quad \text{\large\textasciicircum}\[1em]
\end{align}
$$
$\color{steelblue}\text{Scientific Notation}$ $$
\begin{align}
\text{Sign: } &\quad \pm\[1em]
\text{Significant digits: } &\quad a.bc\[1em]
\text{Exponent: } &\quad e\[1em]
\text{Base: } &\quad 10\[1em]
\end{align}
$$
$\color{steelblue}\text{IEEE Half-Precision}$ $$
\begin{align}
\overbrace{0}^s &\overbrace{10101}^e \overbrace{1010101010}^f\[2em]
\text{Sign: } &\quad s (1-bit)\[1em]
\text{Exponent: } &\quad e (5 bits)\[1em]
\text{Significant digits: } &\quad f \text{(10 bits dropping leading 1)}\[1em]
\end{align}
$$
$\color{steelblue}\text{Recursion}$ Factorial function on $\Bbb{N}$ defined as:
$$
\begin{align}
n! = \stackrel{n}{\prod_{\substack{j=1}}}j
&= 1\cdot 2 \cdots (n-1) \cdot n
\end{align}
$$
Also $n$ ! recursively:
$$
\begin{align}
n! &= \begin{cases}
1 &: n=1 \\
n(n-1)! &: n>1
\end{cases}
\end{align}
$$
Fibonacci sequence:
$$
\begin{align}
f(n) &= \begin{cases}
1 &: n=1 \\
1 &: n=2 \\
f(n-1)+f(n-2) &: n>2
\end{cases}
\end{align}
$$
$\color{steelblue}\text{Python}$ """
MODULO OPERATOR
"""
print (17 % 4 ) # Prints 1 to the console
# 1
"""
EXPONENTS AND LOGARITHMS
"""
> >> import math # Must import math library
> >> math .log2 (8 ) # log2 means log base 2 and returns a float (decimal)
# 3.0
> >> 2 ** 3 # ** is exponentiation
# 8
> >> math .log2 (100 ) / math .log2 (10 ) # base transformation
# 2.0
"""
UFT-8
"""
> >> ord ('A' ) # Convert character to UTF code point
# 65
> >> chr (65 ) # Convert UTF code point to character
# 'A'
"""
HEXADECIMAL
"""
> >> hex (65532 ) # Hex string from integer
# '0xfffc'
> >> "The number is {:x}" .format (65532 ) # Alternative method
# 'The number is fffc'
> >> 0xfffc # Hex literal integer
# 65532
"""
NUMBERS
"""
> >> 9 / 2 # Regular division always returns float
# 4.5
> >> 9 // 2 # Floor division returns integer
# 4
"""
FIBONACCI
"""
def F (n ):
if n == 1 : return 1
elif n == 2 : return 1
else : return F (n - 1 ) + F (n - 2 )
"""
SET THEORY
"""
> >> S = {1 ,2 ,3 }; T = {1 ,3 ,5 } # Braces define sets
> >> S .add (4 ); print (S ) # Sets are mutable
# {1, 2, 3, 4}
> >> S .remove (4 ); print (S )
# {1, 2, 3}
> >> 1 in S # Testing membership
# True
> >> 4 in S
# False
> >> S .issubset ({1 ,2 ,3 ,4 ,5 }) # Testing subset
# True
> >> S <= {1 ,2 ,3 ,4 ,5 } # Alternative subset test
# True
> >> S .union (T ) # Using union
# {1, 2, 3, 5}
> >> S | T # Alternative union
# {1, 2, 3, 5}
> >> S .union (T , {8 ,9 }, {10 ,11 }) # Multiple union
# {1, 2, 3, 5, 8, 9, 10, 11}
> >> someSets = [S , T , {8 ,9 }, {10 ,11 }]
> >> set .union (* someSets ) # Splat operator
# {1, 2, 3, 5, 8, 9, 10, 11}
> >> S .intersection (T ) # Splat and multi-sets would also work
# {1, 3}
> >> S & T # Alternative intersection
# {1, 3}
> >> S - T # Testing difference
# {2}
> >> S .isdisjoint (T ) # is S & T empty?
# False
> >> len (S ) # Size of S
# 3
# Setbuilder notation combines of comprehension and replacement:
> >> S = {0 , 2 , 4 , 6 , 8 }
> >> def p (x ): return x % 2 == 0
...
> >> def p (x ): return x * 5
...
> >> { f (x ) for x in S if p (x ) }
# {0, 40, 10, 20, 30}
> >> { s * 5 for s in range (0 , 10 ) if s % 2 == 0 }
# {0, 40, 10, 20, 30}
"""
PREDICATES AND QUANTIFIERS
"""
> >> def p (x ,y ): # Any function that returns T/F to be used as predicate
return x >= y
> >> p (2 ,1 )
# True
> >> def p (x ): return x >= 0
> >> S = { - 1 , 0 , 1 }; T = { 0 , 1 , 2 }
> >> Sp = [ p (x ) for x in S ] # List of booleans
> >> Tp = [ p (x ) for x in T ] # List of booleans
> >> all (Tp ) # Check for all: ALL True
# True
> >> all (Sp ) # Check there exists: at least ONE True
# True
> >> any (p (x ) for x in S ) # Generator expression
# True