Chapter 10 Symbols

10.1 Greek symbols

10.1.1 Greek symbols

\[ \begin {array}{lcc} \hline \bf symbol & \bf small & \bf capital \\ \hline \text {alpha} & \alpha & A\\ \text {beta} & \beta & B\\ \text {gamma} & \gamma & \Gamma \\ \text {delta} & \delta & \Delta \\ \text {e-psilon} & \epsilon & E\\ \text {zeta} & \zeta & Z\\ \text {eta} & \eta & H\\ \text {theta} & \theta \text { or } \vartheta & \Theta \\ \text {jota} & \iota & I\\ \text {kappa} & \kappa & K\\ \text {lambda} & \lambda & \Lambda \\ \text {mu} & \mu & M\\ \text {nu} & \nu & N\\ \text {ksi} & \xi & \Xi \\ \text {o-mikron} & o & O\\ \text {pi} & \pi \text { or } \varpi & \Pi \\ \text {ro} & \rho \text { or } \varrho & P\\ \text {sigma} & \sigma \text { or } \varsigma & \Sigma \\ \text {tau} & \tau & T\\ \text {u-psilon} & \upsilon & \Upsilon \\ \text {fi} & \phi \text { or } \varphi & \Phi \\ \text {chi} & \chi & X\\ \text {psi} & \psi & \Psi \\ \text {o-mega} & \omega & \Omega \\ \hline \end {array} \]

10.2 Logic Symbols

10.2.1 Propositional calculus

\[ \begin {array}{c|l|l} \bf symbol & \bf name & \bf description \\ \hline \Rightarrow & \text {implication} & \text {if ... then also ...} \\ & & \text {implies} \\ \Leftrightarrow & \text {equivalence} & \text {is equivalent with} \\ \wedge & \text {conjunction} & \text {and} \\ \vee & \text {disjunction} & \text {or} \\ \dot {\vee } & \text {excluding or} & \text {either} \\ \neg & \text {negation} & \text {not} \\ \end {array} \]

10.2.2 Predicate logic

\[ \begin {array}{c|l|l} \bf symbol & \bf name & \bf discription \\ \hline \forall & \text {universal quantification} & \text {for all} \\ \exists & \text {existential quantification} & \text {there exists} \\ \exists ! & & \text {there is precisely one} \\ \nexists & & \text {for no} \\ \end {array} \]

10.3 Sets

10.3.1 Number systems

\[ \begin {array}{rl} \hline \bf set & \bf symbol\\ \hline \text {natural numbers} & \mathbb {N}\\ \text {integers} & \mathbb {Z}\\ \text {rational numbers} & \mathbb {Q}\\ \text {real numbers} & \mathbb {R}\\ \text {complex numbers} & \mathbb {C}\\ \text {quaternions} & \mathbb {H}\\ \end {array} \]

Remark: \(\mathbb {N} \subset \mathbb {Z} \subset \mathbb {Q} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H}\)

10.3.2 Set theory

\[ \begin {array}{c|l|l} \bf symbol & \bf name & \bf definition \\ \hline \{...\} & \text {set} & \text {the collection of ...} \\ \{...|...\} & \text {set} & \text {the set of ... such that ...} \\ \emptyset \text { of } \{\} & \text {empty} & \text {the empty set} \\ \in & \text {membership} & \text {element in} \\ \subseteq & \text {subset} & \text {subset of} \\ \cup & \text {union} & \text {union of ... and ...} \\ \cap & \text {intersection} & \text {intersection of ... and ...}\\ \setminus & \text {set difference} & \text {minus}\\ \times & \text {cartesian product} & \text {times}\\ 2^X & \text {power set} & \text {set of all subsets}\\ \# & \text {cardinality} & \text {number of elements} \end {array} \]

10.4 Mathematical constants

10.4.1 Pi

Ratio of a circle’s circumference to the diameter

\[\pi = 3.14159\, 26535\, 89793\, 23846\, ...\]

10.4.2 Euler’s number

The limit \(e = \lim _{n \to +\infty }\left (1+\dfrac {1}{n}\right )^n\)

\[e = 2.71828\, 18284\, 59045\, 23536\, ...\]

10.4.3 Pythagoras constant

Positive root of \(x^2 = 2\)

\[\sqrt {2} = 1.41421\, 35623\, 73095\, 04880\, ...\]

10.4.4 Theodorus constant

Positive root of \(x^2 = 3\)

\[\sqrt {3} = 1.73205\, 08075\, 68877\, 29352\, ...\]

10.4.5 Golden ratio (Phi)

The ratio of two numbers such that the ratio of their sum to the larger of the two numbers is the same, thus \(\varphi = \dfrac {a}{b} = \dfrac {a+b}{a}= \dfrac {1+\sqrt {5}}{2}\)

\[\varphi = 1.61803\, 39887\, 49894\, 84820\, ...\]

10.4.6 Imaginary unit

The number \(i\in \mathbb {C}\) is any root of \(x^2 = -1\).