Year: 2019

Introductory Sage Tutorial — PREP Tutorials v8.9

Jovan Stosic October 14, 2019

Annotating with Sage

Whether one uses Sage in the classroom or in research, it is usually helpful to describe to the reader what is being done, such as in the description you are now reading.

Jupyter Annotation

Thanks to a styling language called Markdown and the TeX rendering engine called MathJax, you can type much more in Sage than just Sage commands. This math-aware setup makes Sage perfect for annotating computations.

While the Jupyter notebook does not have as fully-featured a word processor as the SageNB, we can still do a fair amount. To use this functionality, we create a Markdown cell (as opposed to a input cell that contains Sage commands that Sage evaluates).

To do this without the keyboard shortcut, there is a menu for each cell; select “Markdown”.

Source: Introductory Sage Tutorial — PREP Tutorials v8.9

Introductory Sage Tutorial — PREP Tutorials v8.9 was last modified: October 14th, 2019 by Jovan Stosic

List of LaTeX mathematical symbols

Jovan Stosic October 13, 2019

Greek letters

Greek letters
Symbol LaTeX Symbol LaTeX
A {\displaystyle \mathrm {A} } and α {\displaystyle \alpha } \Alpha and \alpha N {\displaystyle \mathrm {N} } and ν {\displaystyle \nu } \Nu and \nu
B {\displaystyle \mathrm {B} } and β {\displaystyle \beta } \Beta and \beta Ξ {\displaystyle \Xi } and ξ {\displaystyle \xi } \Xi and \xi
Γ {\displaystyle \Gamma } and γ {\displaystyle \gamma } \Gamma and \gamma O {\displaystyle \mathrm {O} } and o {\displaystyle \mathrm {o} } \Omicron and \omicron
Δ {\displaystyle \Delta } and δ {\displaystyle \delta } \Delta and \delta Π {\displaystyle \Pi } , π {\displaystyle \pi } and ϖ {\displaystyle \varpi } \Pi, \pi and \varpi
E {\displaystyle \mathrm {E} } , ϵ {\displaystyle \epsilon } and ε {\displaystyle \varepsilon } \Epsilon, \epsilon and \varepsilon P {\displaystyle \mathrm {P} } , ρ {\displaystyle \rho } and ϱ {\displaystyle \varrho } \Rho, \rho and \varrho
Z {\displaystyle \mathrm {Z} } and ζ {\displaystyle \zeta } \Zeta and \zeta Σ {\displaystyle \Sigma } , σ {\displaystyle \sigma } and ς {\displaystyle \varsigma \,} \Sigma, \sigma and \varsigma
H {\displaystyle \mathrm {H} } and η {\displaystyle \eta } \Eta and \eta T {\displaystyle \mathrm {T} } and τ {\displaystyle \tau } \Tau and \tau
Θ {\displaystyle \Theta } , θ {\displaystyle \theta } and ϑ {\displaystyle \vartheta } \Theta, \theta and \vartheta Υ {\displaystyle \Upsilon } and υ {\displaystyle \upsilon } \Upsilon and \upsilon
I {\displaystyle \mathrm {I} } and ι {\displaystyle \iota } \Iota and \iota Φ {\displaystyle \Phi } , ϕ {\displaystyle \phi } , and φ {\displaystyle \varphi } \Phi, \phi and \varphi
K {\displaystyle \mathrm {K} } , κ {\displaystyle \kappa } and ϰ {\displaystyle \varkappa } \Kappa, \kappa and \varkappa X {\displaystyle \mathrm {X} } and χ {\displaystyle \chi } \Chi and \chi
Λ {\displaystyle \Lambda } and λ {\displaystyle \lambda } \Lambda and \lambda Ψ {\displaystyle \Psi } and ψ {\displaystyle \psi } \Psi and \psi
M {\displaystyle \mathrm {M} } and μ {\displaystyle \mu } \Mu and \mu Ω {\displaystyle \Omega } and ω {\displaystyle \omega } \Omega and \omega
Archaic Greek letters
Symbol LaTeX
Ϝ {\displaystyle \mathrm {\Digamma} } \Digamma
ϝ {\displaystyle \digamma } \digamma

Unary operators

Unary operators
Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment
+ {\displaystyle +\,} + − {\displaystyle -\,} - negation ! {\displaystyle !\,} ! factorial # {\displaystyle \#} \# primorial
¬ {\displaystyle \neg } \neg not

Relation operators

Relation operators
Symbol LaTeX Comment Symbol LaTeX Comment
< {\displaystyle <\,} < is less than > {\displaystyle >\,} > is greater than
≮ {\displaystyle \nless } \nless is not less than ≯ {\displaystyle \ngtr } \ngtr is not greater than
≤ {\displaystyle \leq } \leq is less than or equal to ≥ {\displaystyle \geq } \geq is greater than or equal to
⩽ {\displaystyle \leqslant } \leqslant is less than or equal to ⩾ {\displaystyle \geqslant } \geqslant is greater than or equal to
≰ {\displaystyle \nleq } \nleq is neither less than nor equal to ≱ {\displaystyle \ngeq } \ngeq is neither greater than nor equal to
⪇ {\displaystyle \nleqslant } \nleqslant is neither less than nor equal to ⪈ {\displaystyle \ngeqslant } \ngeqslant is neither greater than nor equal to
≺ {\displaystyle \prec } \prec precedes ≻ {\displaystyle \succ } \succ succeeds
⊀ {\displaystyle \nprec } \nprec doesn’t precede ⊁ {\displaystyle \nsucc } \nsucc doesn’t succeed
⪯ {\displaystyle \preceq } \preceq precedes or equals ⪰ {\displaystyle \succeq } \succeq succeeds or equals
⋠ {\displaystyle \npreceq } \npreceq neither precedes nor equals ⋡ {\displaystyle \nsucceq } \nsucceq neither succeeds nor equals
≪ {\displaystyle \ll } \ll ≫ {\displaystyle \gg } \gg
⋘ {\displaystyle \lll } \lll ⋙ {\displaystyle \ggg } \ggg
⊂ {\displaystyle \subset } \subset is a proper subset of ⊃ {\displaystyle \supset } \supset is a proper superset of
⊄ {\displaystyle \not \subset } \not\subset is not a proper subset of ⊅ {\displaystyle \not \supset } \not\supset is not a proper superset of
⊆ {\displaystyle \subseteq } \subseteq is a subset of ⊇ {\displaystyle \supseteq } \supseteq is a superset of
⊈ {\displaystyle \nsubseteq } \nsubseteq is not a subset of ⊉ {\displaystyle \nsupseteq } \nsupseteq is not a superset of
⊏ {\displaystyle \sqsubset } \sqsubset ⊐ {\displaystyle \sqsupset } \sqsupset
⊑ {\displaystyle \sqsubseteq } \sqsubseteq ⊒ {\displaystyle \sqsupseteq } \sqsupseteq
Symbol LaTeX Comment
= {\displaystyle =\,} = is equal to
≐ {\displaystyle \doteq } \doteq
≡ {\displaystyle \equiv } \equiv is equivalent to
≈ {\displaystyle \approx } \approx is approximately
≅ {\displaystyle \cong } \cong is congruent to
≃ {\displaystyle \simeq } \simeq is similar or equal to
∼ {\displaystyle \sim } \sim is similar to
∝ {\displaystyle \propto } \propto is proportional to
≠ {\displaystyle \neq } or ≠ {\displaystyle \neq } \neq or \ne is not equal to
Symbol LaTeX Comment Symbol LaTeX Comment
∥ {\displaystyle \parallel } \parallel is parallel with ∦ {\displaystyle \nparallel } \nparallel is not parallel with
≍ {\displaystyle \asymp } \asymp is asymptotic to ⋈ {\displaystyle \bowtie } \bowtie
⊢ {\displaystyle \vdash } \vdash ⊣ {\displaystyle \dashv } \dashv
∈ {\displaystyle \in } \in is member of ∋ {\displaystyle \ni } \ni owns, has member
⌣ {\displaystyle \smile } \smile ⌢ {\displaystyle \frown } \frown
⊨ {\displaystyle \models } \models models ∉ {\displaystyle \notin } \notin is not member of
⊥ {\displaystyle \perp } \perp is perpendicular with ∣ {\displaystyle \mid } \mid divides

Binary operators

Binary operators
Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment
± {\displaystyle \pm } \pm plus or minus ∩ {\displaystyle \cap } \cap set intersection ⋄ {\displaystyle \diamond } \diamond ⊕ {\displaystyle \oplus } \oplus
∓ {\displaystyle \mp } \mp minus or plus ∪ {\displaystyle \cup } \cup set union △ {\displaystyle \bigtriangleup } \bigtriangleup ⊖ {\displaystyle \ominus } \ominus
× {\displaystyle \times } \times multiplied by ⊎ {\displaystyle \uplus } \uplus multiset addition ▽ {\displaystyle \bigtriangledown } \bigtriangledown ⊗ {\displaystyle \otimes } \otimes
÷ {\displaystyle \div } \div divided by ⊓ {\displaystyle \sqcap } \sqcap ◃ {\displaystyle \triangleleft } \triangleleft ⊘ {\displaystyle \oslash } \oslash
∗ {\displaystyle \ast } \ast asterisk ⊔ {\displaystyle \sqcup } \sqcup ▹ {\displaystyle \triangleright } \triangleright ⊙ {\displaystyle \odot } \odot
⋆ {\displaystyle \star } \star ∨ {\displaystyle \vee } \vee ◯ {\displaystyle \bigcirc } \bigcirc ∘ {\displaystyle \circ } \circ
† {\displaystyle \dagger } \dagger ∧ {\displaystyle \wedge } \wedge ∙ {\displaystyle \bullet } \bullet ∖ {\displaystyle \setminus } \setminus set difference
‡ {\displaystyle \ddagger } \ddagger ⋅ {\displaystyle \cdot } \cdot ≀ {\displaystyle \wr } \wr ⨿ {\displaystyle \amalg } \amalg

Negated binary relations

Negated binary operators
Symbol LaTeX Comment Symbol LaTeX Comment
≠ {\displaystyle \neq } or ≠ {\displaystyle \neq \,} \neq or \ne is not equal to ∉ {\displaystyle \notin } \notin is not member of
≮ {\displaystyle \nless } \nless is not less than ≯ {\displaystyle \ngtr } \ngtr is not greater than
≰ {\displaystyle \nleq } \nleq is not less than or equal to ≱ {\displaystyle \ngeq } \ngeq is not greater than or equal to
⪇ {\displaystyle \nleqslant } \nleqslant ⪈ {\displaystyle \ngeqslant } \ngeqslant
≰ {\displaystyle \nleqq } \nleqq ≱ {\displaystyle \ngeqq } \ngeqq
⪇ {\displaystyle \lneq } \lneq ⪈ {\displaystyle \gneq } \gneq
≨ {\displaystyle \lneqq } \lneqq ≩ {\displaystyle \gneqq } \gneqq
≨ {\displaystyle \lvertneqq } \lvertneqq ≩ {\displaystyle \gvertneqq } \gvertneqq
⋦ {\displaystyle \lnsim } \lnsim ⋧ {\displaystyle \gnsim } \gnsim
⪉ {\displaystyle \lnapprox } \lnapprox ⪊ {\displaystyle \gnapprox } \gnapprox
⊀ {\displaystyle \nprec } \nprec does not precede ⊁ {\displaystyle \nsucc } \nsucc does not succeed
⋠ {\displaystyle \npreceq } \npreceq neither precedes nor equals ⋡ {\displaystyle \nsucceq } \nsucceq neither succedes nor equals
⪵ {\displaystyle \precneqq } \precneqq ⪶ {\displaystyle \succneqq } \succneqq
⋨ {\displaystyle \precnsim } \precnsim ⋩ {\displaystyle \succnsim } \succnsim
⪹ {\displaystyle \precnapprox } \precnapprox ⪺ {\displaystyle \succnapprox } \succnapprox
≁ {\displaystyle \nsim } \nsim is not similar to ≆ {\displaystyle \ncong } \ncong is not congruent to
∤ {\displaystyle \nshortmid } \nshortmid ∦ {\displaystyle \nshortparallel } \nshortparallel
∤ {\displaystyle \nmid } \nmid ∦ {\displaystyle \nparallel } \nparallel is not parallel with
⊬ {\displaystyle \nvdash } \nvdash ⊭ {\displaystyle \nvDash } \nvDash
⊮ {\displaystyle \nVdash } \nVdash ⊯ {\displaystyle \nVDash } \nVDash
⋪ {\displaystyle \ntriangleleft } \ntriangleleft ⋫ {\displaystyle \ntriangleright } \ntriangleright
⋬ {\displaystyle \ntrianglelefteq } \ntrianglelefteq ⋭ {\displaystyle \ntrianglerighteq } \ntrianglerighteq
⊈ {\displaystyle \nsubseteq } \nsubseteq ⊉ {\displaystyle \nsupseteq } \nsupseteq
⊈ {\displaystyle \nsubseteqq } \nsubseteqq ⊉ {\displaystyle \nsupseteqq } \nsupseteqq
⊊ {\displaystyle \subsetneq } \subsetneq ⊋ {\displaystyle \supsetneq } \supsetneq
⊊ {\displaystyle \varsubsetneq } \varsubsetneq ⊋ {\displaystyle \varsupsetneq } \varsupsetneq
⫋ {\displaystyle \subsetneqq } \subsetneqq ⫌ {\displaystyle \supsetneqq } \supsetneqq
⫋ {\displaystyle \varsubsetneqq } \varsubsetneqq ⫌ {\displaystyle \varsupsetneqq } \varsupsetneqq

Set and/or logic notation

Set notation
Symbol LaTeX Comment
∅ {\displaystyle \emptyset } or ∅ {\displaystyle \emptyset \,} , and ∅ {\displaystyle \varnothing \,} \O or \emptyset, and \varnothing the empty set
N {\displaystyle \mathbb {N} } \N set of natural numbers
Z {\displaystyle \mathbb {Z} } \Z set of integers
Q {\displaystyle \mathbb {Q} } \Q set of rational numbers
A {\displaystyle \mathbb {A} } \mathbb{A} set of algebraic numbers
R {\displaystyle \mathbb {R} } \R set of real numbers
C {\displaystyle \mathbb {C} } \C set of complex numbers
H {\displaystyle \mathbb {H} } \mathbb{H} set of quaternions
O {\displaystyle \mathbb {O} } \mathbb{O} set of octonions
S {\displaystyle \mathbb {S} } \mathbb{S} set of sedenions
∈ {\displaystyle \in } \in is member of
∉ {\displaystyle \notin } \notin is not member of
∋ {\displaystyle \ni } \ni owns (has member)
⊂ {\displaystyle \subset } \subset is proper subset of
⊆ {\displaystyle \subseteq } \subseteq is subset of
⊃ {\displaystyle \supset } \supset is proper superset of
⊇ {\displaystyle \supseteq } \supseteq is superset of
∪ {\displaystyle \cup } \cup set union
∩ {\displaystyle \cap } \cap set intersection
∖ {\displaystyle \setminus } \setminus set difference
Logic notation
Symbol LaTeX Comment
∃ {\displaystyle \exists } \exists there exists at least one
∃ ! {\displaystyle \exists !} \exists! there exists one and only one
∄ {\displaystyle \nexists } \nexists there is no
∀ {\displaystyle \forall } \forall for all
¬ {\displaystyle \neg } \neg not (logical not)
∨ {\displaystyle \lor } \lor or (logical or)
∧ {\displaystyle \land } \land and (logical and)
⟹ {\displaystyle \Longrightarrow } or ⟹ {\displaystyle \implies } \Longrightarrow or \implies implies
⇒ {\displaystyle \Rightarrow } \Rightarrow (preferred for right implication)
⟸ {\displaystyle \Longleftarrow } \Longleftarrow is implied by (only if)
⇐ {\displaystyle \Leftarrow } \Leftarrow (preferred for left implication)
⟺ {\displaystyle \iff } \iff is equivalent to (if and only if, iff)
⇔ {\displaystyle \Leftrightarrow } \Leftrightarrow (preferred for equivalence)
⊤ {\displaystyle \top } \top
⊥ {\displaystyle \bot } \bot

Geometry

Geometry notation
Symbol LaTeX Comment Symbol LaTeX Comment
A B ¯ {\displaystyle {\overline {\rm {AB}}}} \overline{\rm AB} segment A B → {\displaystyle {\overrightarrow {\rm {AB}}}} \overrightarrow{\rm AB} ray (half-line)
∠ {\displaystyle \angle } \angle angle ∡ {\displaystyle \measuredangle } \measuredangle measured angle
△ {\displaystyle \triangle } \triangle triangle ◻ {\displaystyle \square } \square square
≅ {\displaystyle \cong } \cong congruent (same shape and size) ≆ {\displaystyle \ncong } \ncong not congruent
∼ {\displaystyle \sim } \sim similar (same shape) ≁ {\displaystyle \nsim } \nsim not similar
‖ {\displaystyle \|} \| is parallel with ∦ {\displaystyle \nparallel } \nparallel is not parallel with
⊥ {\displaystyle \perp } \perp is perpendicular to ⊥̸ {\displaystyle \not \perp } \not\perp is not perpendicular to

Delimiters

Delimiters
Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment
| {\displaystyle |\,} | divides ‖ {\displaystyle \|} \| divides unitarily, is parallel with / {\displaystyle /\,} / slash ∖ {\displaystyle \backslash } \backslash
( {\displaystyle (\,} ( \, left parenthesis ) {\displaystyle )\,} ) \, right parenthesis [ {\displaystyle [\,} [ \, left [square] bracket ] {\displaystyle ]\,} ] \, right [square] bracket
{ {\displaystyle \{} \{ left brace } {\displaystyle \}} \} right brace ⟨ {\displaystyle \langle } \langle left angle bracket ⟩ {\displaystyle \rangle } \rangle right angle bracket
⌈ {\displaystyle \lceil } \lceil ceiling (left) ⌉ {\displaystyle \rceil } \rceil ceiling (right) ⌊ {\displaystyle \lfloor } \lfloor floor (left) ⌋ {\displaystyle \rfloor } \rfloor floor (right)
⌜ {\displaystyle \ulcorner } \ulcorner ⌝ {\displaystyle \urcorner } \urcorner ⌞ {\displaystyle \llcorner } \llcorner ⌟ {\displaystyle \lrcorner } \lrcorner

Arrows

Arrows
Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment
→ {\displaystyle \rightarrow } or → {\displaystyle \to } \rightarrow or \to ⇒ {\displaystyle \Rightarrow } \Rightarrow ⟶ {\displaystyle \longrightarrow } \longrightarrow ⟹ {\displaystyle \Longrightarrow } \Longrightarrow
↦ {\displaystyle \mapsto } \mapsto ⟼ {\displaystyle \longmapsto } \longmapsto
← {\displaystyle \leftarrow } or ← {\displaystyle \gets } \leftarrow or \gets ⇐ {\displaystyle \Leftarrow } \Leftarrow ⟵ {\displaystyle \longleftarrow } \longleftarrow ⟸ {\displaystyle \Longleftarrow } \Longleftarrow
Symbol LaTeX Comment Symbol LaTeX Comment
↑ {\displaystyle \uparrow } \uparrow Knuth’s up-arrow notation ⇑ {\displaystyle \Uparrow } \Uparrow
↓ {\displaystyle \downarrow } \downarrow ⇓ {\displaystyle \Downarrow } \Downarrow
↕ {\displaystyle \updownarrow } \updownarrow ⇕ {\displaystyle \Updownarrow } \Updownarrow

Other symbols

Other symbols
Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment Symbol LaTeX Comment
∂ {\displaystyle \partial } \partial partial derivative ı {\displaystyle \imath } \imath ℜ {\displaystyle \Re } \Re real part ∇ {\displaystyle \nabla } \nabla del (vector calculus)
ð {\displaystyle \eth } \eth ȷ {\displaystyle \jmath } \jmath ℑ {\displaystyle \Im } \Im imaginary part ◻ {\displaystyle \Box } \Box
ℏ {\displaystyle \hbar } \hbar reduced Planck’s constant ℓ {\displaystyle \ell } \ell ℘ {\displaystyle \wp } \wp [Weierstrass] powerset ∞ {\displaystyle \infty } \infty infinity

 

Hebrew lettters
Symbol LaTeX Comment
ℵ {\displaystyle \aleph } \aleph aleph numbers
ℶ {\displaystyle \beth } \beth
ℷ {\displaystyle \gimel } \gimel

Trigonometric functions

Circular functions
The prefix arc used for inverse circular trigonometric functions is the abbreviation for arcus.
Symbol LaTeX Symbol LaTeX Symbol LaTeX Symbol LaTeX
sin {\displaystyle \sin } \sin arcsin {\displaystyle \arcsin } \arcsin csc {\displaystyle \csc } \csc arccsc {\displaystyle \operatorname {arccsc} } \arccsc
cos {\displaystyle \cos } \cos arccos {\displaystyle \arccos } \arccos sec {\displaystyle \sec } \sec arcsec {\displaystyle \operatorname {arcsec} } \arcsec
tan {\displaystyle \tan } \tan arctan {\displaystyle \arctan } \arctan cot {\displaystyle \cot } \cot arccot {\displaystyle \operatorname {arccot} } \arccot

 

Hyperbolic functions
The abbreviations arcsinh, arccosh, etc., are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area.
Symbol LaTeX Symbol LaTeX Symbol LaTeX Symbol LaTeX
sinh {\displaystyle \sinh } \sinh arsinh {\displaystyle \operatorname {arsinh} } \operatorname{arsinh} csch {\displaystyle \operatorname {csch} } \operatorname{csch} arcsch {\displaystyle \operatorname {arcsch} } \operatorname{arcsch}
cosh {\displaystyle \cosh } \cosh arcosh {\displaystyle \operatorname {arcosh} } \operatorname{arcosh} sech {\displaystyle \operatorname {sech} } \operatorname{sech} arsech {\displaystyle \operatorname {arsech} } \operatorname{arsech}
tanh {\displaystyle \tanh } \tanh artanh {\displaystyle \operatorname {artanh} } \operatorname{artanh} coth {\displaystyle \coth } \coth arcoth {\displaystyle \operatorname {arcoth} } \operatorname{arcoth}

 

 

List of LaTeX mathematical symbols was last modified: October 13th, 2019 by Jovan Stosic

Q.E.D. – Wikipedia

Jovan Stosic October 13, 2019

Q.E.D. or QED (sometimes italicized) is an initialism of the Latin phrase “quod erat demonstrandum”, literally meaning “what was to be shown”.[1] Traditionally, the abbreviation is placed at the end of a mathematical proof or philosophical argument to indicate that the proof or argument is complete, therefore used with the meaning “thus it has been demonstrated”.

Source: Q.E.D. – Wikipedia

Q.E.D. – Wikipedia was last modified: October 13th, 2019 by Jovan Stosic