Letters

Symbol	Description
$ x, y, z $	Usually refers to numeric variables or constants
$ i, j, k $	Used to refer to variables representing indices in a summation, e.g. $ \sum_{i=0}^n i^2 $
$ A, B, C, \dots $	May refer to particular sets (e.g. $S \subset T$), propositional constants (e.g. $A \wedge B$), functions (e.g. $X(x)$), predicates (e.g. $P(X)$), or words in a rewriting system
$ p, q, r, \dots $	Usually refers to propositional variables
$ \Phi, \chi, \Psi $	Greek letters Phi, Chi and Psi; often axiomatic letters used to stand in for well-formed formulas of predicate
$ \Phi(x) $	Often used when talking about some predicate $\Phi$
$ \Sigma $	Used as the symbol for summation or for a set that is an alphabet of symbols
$ \Pi $	Used as the symbol for product
$ \mathbb{N},\mathbb{Q},\mathbb{R}, \dots $	Blackboard-style letters usually refer to particular types; see below.
$ \Omega $	Probability sample space
$ P(X) $	Probability function
$ \sigma, s $	Standard deviation (population and sample)
$ \mu $	Mean (population mean)
$ \overline{x} $	Mean of variable $x$
$ z, t, F, \chi^2 $	Particular inferrential test statistics
$ p $	P-value, probability a positive inferrential test result would be observed by chance
$ \alpha $	Alpha, used in inferrential statistics as a threshold for p

Types

Symbol	Elements
$ \mathbb{N} $	$ \{ 1, 2, 3, 4, 5, \dots \} $
$ \mathbb{R} $	$ \{ \dots, -0.001020\dots, e, \pi,\dots \} $
$ \mathbb{Q} $	$ \{ \dots, \frac{1}{1}, \frac{1}{2}, \frac{2}{3}, \dots \} $
$ \{ 0, 1 \} $	$ \{ 0, 1 \} $
$ \Sigma^* $	$ \{a, b, abc, aabcbdc, \dots \} $

Rewriting Systems

Name	Symbol	Example
Alphabet	$ \Sigma $	$ \Sigma = \{a, b, c\} $
Set of words on $ \Sigma $	$ \Sigma^* $	$ \Sigma^* = \{\epsilon, a, aa, b, bb, c, cc, \dots \} $
Empty string	$ \epsilon $	$ \epsilon \in \Sigma^* $
Transition	$ \Rightarrow $	$ A \Rightarrow B $
Derivation	$ \Rightarrow^* $	$ A \Rightarrow^* B $
Rewritten-as	$ \curvearrowright $	$ X \curvearrowright aXb $

Propositional Logic

Name	Symbol
True	$ \top, 1, \text{true} $
False	$ \bot, 0, \text{false} $

Propositional Logic Operators

Name	Symbol	Example
Negation	$\neg$	$ \neg \top = \bot $
Conjunction	$\wedge$	$ \top \wedge \bot = \bot $
Disjunction	$\vee$	$ \top \wedge \bot = \top $
Implication	$\implies$	$ \bot \implies \bot = \top $
Equivalence	$\Longleftrightarrow$	$ \top \Longleftrightarrow \top = \top $

Set Theory

Set Relations

Name	Symbol	Example
element of	$\in$	$ a \in \{ a \} $
subset	$\subseteq$	$ \{ a \} \subseteq \{ a, b \} $
superset	$\supseteq$	$ \{ a, b \} \supseteq \{ a, b \} $
equality	$=$	$ \{ a \} = \{ a \} $
proper subset	$\subset$	$ \{ a \} \subset \{ a, b \} $
proper superset	$\supset$	$ \{ a, b \} \supset \{ a \} $

Set Operations

Name	Symbol	Example
union	$\cup$	$ \{ a \} \cup \{ b \} = \{ a, b \} $
intersection	$\cap$	$ \{ a, b \} \cap \{ b, c \} = \{ b \} $
set difference	$\setminus$	$ \{ a, b \} \setminus \{ b \} = \{ a \} $
cardinality	$\#$	$ \#(\{ a, b \}) = 2 $
power set	$ \mathbb{P} $	$ \mathbb{P}(\{ a, b \} = \{ \emptyset, \{a\}, \{b\}, \{a, b\} \} $
cartesian product	$\times$	$ \{ a \} \times \{ 1 \} = \{(a, 1), (a, 2)\} $

Predicate Logic

Quantifiers

Name	Symbol	Example
Existential	$ \exists $	$ (\exists x \in \mathbb{N})(x^2 = x) $
Universal	$ \forall $	$ (\forall x \in \mathbb{N})(x > 0) $

Set Comprehensions

Name	Symbol	Example
Term	$ \bullet $	$ \{ x : \mathbb{N} \bullet 2x \} = \{2, 4, 6, 8, \dots \} $
Such-that	$ \mid $	$ \{ x : \mathbb{N} \| x \mod 2 = 0 \} = \{2, 4, 6, 8, \dots \} $

Axiom Schema

Propositional Logic Axiom Schema

Law 1.1: Complement Law of Negation

\[ ( \neg \top) \iff \bot \]

\[ ( \neg \bot) \iff \top \]

Law 1.2: Double Negation

\[ ( \neg \neg \phi) \iff \phi \]

Law 2.1: Idempotence of Conjunction

\[ (\phi \wedge \phi) \iff \phi \]

Law 2.2: Conjunction identity

\[ (\phi \wedge \top) \iff \phi \]

Law 2.3: Domination Law of Conjunction

\[ (\phi \wedge \bot) \iff \bot \]

Law 2.4: Complement law

\[ (\phi \wedge ( \neg \phi)) \iff \bot \]

Law 2.5: Commutativity of Conjunction

\[ (\phi \wedge \chi) \iff (\chi \wedge \phi) \]

Law 2.6: Associativity of Conjunction

\[ (\phi \wedge \chi) \wedge \psi \iff \phi \wedge (\chi \wedge \psi) \]

Law 3.1: de Morgan’s Laws

\[ \neg (\phi \wedge \chi) \iff (( \neg \phi) \vee ( \neg \chi)) \]

\[ \neg (\phi \vee \chi) \iff (( \neg \phi) \wedge ( \neg \chi)) \]

Law 3.2: Idempotence of Disjunction

\[ (\phi \vee \phi) \iff \phi \]

Law 3.3: Disjunction Identity

\[ (\phi \vee \bot) \iff \phi \]

Law 3.4: Domination Law of Disjunction

\[ (\phi \vee \top) \iff \top \]

Law 3.5: Associativity of Disjunction

\[ \phi \vee (\chi \vee \psi) \iff (\phi \vee \chi) \vee \psi \]

Law 3.6: Commutativity of Disjunction

\[ \phi \vee \chi \iff \chi \vee \phi \]

Law 3.7: Complement law

\[ (( \neg \phi) \vee \phi) \iff \top \]

Law 3.8: Disjunction distributes through conjunction

\[ \phi \vee (\chi \wedge \psi) \iff (\phi \vee \chi) \wedge (\phi \vee \psi) \]

Law 3.9: Conjunction distributes through disjunction

\[ \phi \wedge (\chi \vee \psi) \iff (\phi \wedge \chi) \vee (\phi \wedge \psi) \]

Law 4.1: Definition of Implication

\[ (\phi \implies \chi) \iff (( \neg \phi) \vee \chi) \]

Law 5.1: Associativity of Equivalence

\[ ((\phi \iff \chi) \iff \psi) \iff (\phi \iff (\chi \iff \psi)) \]

Law 5.2: Commutativity of Equivalence

\[ (\phi \iff \chi) \iff (\chi \iff \phi) \]

Law 5.3: Equivalence Identity

\[ (\phi \iff \phi) \iff \top \]

Law 5.4: Complement Law

\[ (\phi \iff ( \neg \phi)) \iff \bot \]

Law 5.5: Definition of Equivalence

\[ (\phi \iff \chi) \iff ((\phi \implies \chi) \wedge (\chi \implies \phi)) \]

Set Theory Axiom Schema

Law 1.1 for any set S and any element s

\[ \neg(s \in S) \iff s \not\in S \]

Law 1.2 for any element x

\[ x \in \emptyset \iff \text{false} \]

Law 2.1 for any sets S and T

\[ (S \subseteq T \wedge T \subseteq S) \iff S = T \]

Law 2.2 for any sets S

\[ ( \emptyset \subseteq S) \]

Law 2.3 all sets are a subset of themselves

\[ (S \subseteq S) \]

Law 2.4 for any sets S and T

\[ \neg(S \subseteq T) \iff S \not\subseteq T \]

Law 2.5 for any sets S and T

\[ S \subseteq T \iff (S \subset T \vee S = T) \]

Law 2.6 for any sets S and T

\[ S \not\subset T \iff \neg(S \subset T) \]

Law 2.7 for any set S

\[ S \not\subset S \]

Law 2.8 for any sets S and T

\[ S \subset T \implies T\not\subset S \]

Law 3.1 for any sets S and T. Stating S is a superset of T is logically equivalent to stating that T is a subset of S

\[ S \supseteq T \iff T \subseteq S \]

Law 4.1 for any element a, and any sets S and T

\[ a \in S \cup T \iff (a \in S \vee a \in T) \]

Law 4.2 combining Set S with the empty set Ø, is equivalent to Set S:

\[ S \cup \emptyset =S \]

Law 4.3 The set union of any set S combined with itself is equivalent to itself

\[ S \cup S=S \]

Law 4.4 Union is commutative

\[ S \cup T=T \cup S \]

Law 4.5 Union is associative

\[ R \cup (S \cup T) = (R \cup T) \cup S \]

Law 4.6 The union of two sets is always at least as big as each set considered individually

\[ S \subseteq S \cup T \]

Law 5.1 where a given element a is in the intersection of sets S and T is must be an element of both sets

\[ a \in S \cap T \iff (a \in S \wedge a \in T) \]

Law 5.2 the intersection of a given set S with the empty set Ø is always the empty set

\[ S \cap \emptyset = \emptyset \]

Law 5.3 the intersection of set S with itself is always S

\[ S \cap S = S \]

Law 5.4 Intersection is commutative

\[ S \cap T=T \cap S \]

Law 5.5 Intersection is associative

\[ R \cap (S \cap T) = (R \cap S) \cap T \]

Law 5.6 The intersection of any given sets is always at least as small as one of the given sets

\[ S \cap T \subseteq S \]

Law 5.7 union distributes through Intersection and Intersection distributes through distribution

\[ R \cup (S \cap T) = (R \cup S) \cap (R \cup T) \]

\[ R \cap (S \cup T) = (R \cap S) \cup (R \cap T) \]

Law 6.1 if $a$ is an element of the Set difference of Sets $S \setminus T$ then $S$ is a member of the former and not the latter

\[ a \in S \setminus T \iff (a \in S \wedge a \not\in T) \]

Law 6.2 Set S intersected with the empty set is equivocal to set S

\[ S \setminus \emptyset = S \]

Law 6.3 The set difference of the empty set with a set S is the empty set

\[ \emptyset \setminus S = \emptyset \]

Law 6.4 The difference in any set S and itself produces the empty set

\[ S \setminus S = \emptyset \]

Law 6.5 The difference in Set R and the union or sets S and T is equivocal to the union of the difference in set R and S and R and T. A similar property holds for Intersection.

\[ R \setminus (S \cup T) = (R \setminus S) \cap (R \setminus T) \]

\[ R \setminus (S \cap T) = (R \setminus S) \cup (R \setminus T) \]

Law 7.1 When two different sets have exactly the same elements, they are equal

\[ (S = T) \Longleftrightarrow (\forall x ) ( x\in S\iff x \in T ) \]

Law 8.1 the cardinality of the empty set is 0

\[ \# \emptyset = 0 \]

Law 8.2 the cardinality of the intersection of S and T is equal to the cardinality of S minus the cardinality of the set difference of S and T

\[ \#(S \cap T) = \#S - \#(S\setminus T) \]

Law 8.3 the cardinality of the union of S and T is equal to the cardinality of S plus the cardinality of T minus the cardinality of the intersection of S and T

\[ \#(S \cup T) = \#S + \#T - \#(S \cap T) \]

Law 8.4 the Cardinality of the intersection of S and T is equal to the cardinality of S minus the cardinality of the intersection of S and T

\[ \#(S \setminus T) = \#S - \#(S \cap T) \]

Law 8.5 the Cardinality of the cartesian product of S and R is the product of the cardinalities of S and R

\[ \#(S \times T) = \#S \times \#T \]

Law 9.1 Set S is an element of the power set of T if and only if S is a subset of T

\[ S \in \mathbb{P} (T) \iff S \subseteq T \]

Law 9.2 the empty set is an element of the power set of a any given set S

\[ \emptyset \in \mathbb{P} (S) \]

Law 9.3 for any given set S, S is an element of the power set of itself

\[ S \in \mathbb{P} (S) \]

Law 9.4 the cardinality of the power set of set S is equal to two to the power of the cardinality of S

\[ \#( \mathbb{P} (S)) = 2^{\#(S)} \]

Law 9.5 for a given set R which is an element of the power set of S, the intersection of R and S is equal to R.

\[ R \cap S = R \text{, where } R \in \mathbb{P} ( S ) \]

Law 10.1 For a given set T $\subseteq$ S, the compliment of T is equal to the set difference in S and T

\[ T^{-} = S \setminus T \text{, where } T \in \mathbb{P} ( S ) \]

Law 10.2 For a given set T $\subseteq$ S, The union of T and its compliment is equal to the S

\[ T \cup T^{-} = S \text{, where } T \in \mathbb{P} ( S ) \]

Law 10.3 For a given set T $\subseteq$ S, The intersection of T and its compliment is equal to the empty set.

\[ T \cap T^{-} = \emptyset \text{, where } T \in \mathbb{P} ( S ) \]

Law 11.1 for any set of sets A and any element a, $a \in \bigcup A $ if, and only if, there is some set $S \in A $ such that $ a \in S$

\[ a \in \bigcup A \iff \exists S \in A . a \in S \]

Law 11.2 for any set of sets A and any element a, $a \in \bigcap A$ if, and only if, for every set $ S \in A $ it is the case that $ a \in S$

\[ a \in \bigcap A \iff \forall S \in A . a \in S \]

Symbol	Elements
\( \mathbb{N} \)	\( \{ 1, 2, 3, 4, 5, \dots \} \)
\( \mathbb{R} \)	\( \{ \dots, -0.001020\dots, e, \pi,\dots \} \)
\( \mathbb{Q} \)	\( \{ \dots, \frac{1}{1}, \frac{1}{2}, \frac{2}{3}, \dots \} \)
\( \{ 0, 1 \} \)	\( \{ 0, 1 \} \)
\( \Sigma^* \)	\( \{a, b, abc, aabcbdc, \dots \} \)

Name	Symbol	Example
Alphabet	\( \Sigma \)	\( \Sigma = \{a, b, c\} \)
Set of words on \( \Sigma \)	\( \Sigma^* \)	\( \Sigma^* = \{\epsilon, a, aa, b, bb, c, cc, \dots \} \)
Empty string	\( \epsilon \)	\( \epsilon \in \Sigma^* \)
Transition	\( \Rightarrow \)	\( A \Rightarrow B \)
Derivation	\( \Rightarrow^* \)	\( A \Rightarrow^* B \)
Rewritten-as	\( \curvearrowright \)	\( X \curvearrowright aXb \)

Symbol	Description
\( x, y, z \)	Usually refers to numeric variables or constants
\( i, j, k \)	Used to refer to variables representing indices in a summation, e.g. \( \sum_{i=0}^n i^2 \)
\( A, B, C, \dots \)	May refer to particular sets (e.g. $S \subset T$), propositional constants (e.g. $A \wedge B$), functions (e.g. $X(x)$), predicates (e.g. $P(X)$), or words in a rewriting system
\( p, q, r, \dots \)	Usually refers to propositional variables
\( \Phi, \chi, \Psi \)	Greek letters Phi, Chi and Psi; often axiomatic letters used to stand in for well-formed formulas of predicate
\( \Phi(x) \)	Often used when talking about some predicate $\Phi$
\( \Sigma \)	Used as the symbol for summation or for a set that is an alphabet of symbols
\( \Pi \)	Used as the symbol for product
\( \mathbb{N},\mathbb{Q},\mathbb{R}, \dots \)	Blackboard-style letters usually refer to particular types; see below.
\( \Omega \)	Probability sample space
\( P(X) \)	Probability function
\( \sigma, s \)	Standard deviation (population and sample)
\( \mu \)	Mean (population mean)
\( \overline{x} \)	Mean of variable $x$
\( z, t, F, \chi^2 \)	Particular inferrential test statistics
\( p \)	P-value, probability a positive inferrential test result would be observed by chance
\( \alpha \)	Alpha, used in inferrential statistics as a threshold for p

Name	Symbol
True	\( \top, 1, \text{true} \)
False	\( \bot, 0, \text{false} \)

Name	Symbol	Example
Negation	$\neg$	\( \neg \top = \bot \)
Conjunction	$\wedge$	\( \top \wedge \bot = \bot \)
Disjunction	$\vee$	\( \top \wedge \bot = \top \)
Implication	$\implies$	\( \bot \implies \bot = \top \)
Equivalence	$\Longleftrightarrow$	\( \top \Longleftrightarrow \top = \top \)

Name	Symbol	Example
element of	$\in$	\( a \in \{ a \} \)
subset	$\subseteq$	\( \{ a \} \subseteq \{ a, b \} \)
superset	$\supseteq$	\( \{ a, b \} \supseteq \{ a, b \} \)
equality	$=$	\( \{ a \} = \{ a \} \)
proper subset	$\subset$	\( \{ a \} \subset \{ a, b \} \)
proper superset	$\supset$	\( \{ a, b \} \supset \{ a \} \)

Name	Symbol	Example
union	$\cup$	\( \{ a \} \cup \{ b \} = \{ a, b \} \)
intersection	$\cap$	\( \{ a, b \} \cap \{ b, c \} = \{ b \} \)
set difference	$\setminus$	\( \{ a, b \} \setminus \{ b \} = \{ a \} \)
cardinality	$\#$	\( \#(\{ a, b \}) = 2 \)
power set	\( \mathbb{P} \)	\( \mathbb{P}(\{ a, b \} = \{ \emptyset, \{a\}, \{b\}, \{a, b\} \} \)
cartesian product	$\times$	\( \{ a \} \times \{ 1 \} = \{(a, 1), (a, 2)\} \)

Name	Symbol	Example
Existential	\( \exists \)	\( (\exists x \in \mathbb{N})(x^2 = x) \)
Universal	\( \forall \)	\( (\forall x \in \mathbb{N})(x > 0) \)

Name	Symbol	Example
Term	\( \bullet \)	\( \{ x : \mathbb{N} \bullet 2x \} = \{2, 4, 6, 8, \dots \} \)
Such-that	\( \mid \)	\( \{ x : \mathbb{N} \| x \mod 2 = 0 \} = \{2, 4, 6, 8, \dots \} \)