biology: Basic mathematical symbols

Thursday, November 16, 2006

Basic mathematical symbols

Symbol	Name	Explanation	Examples
	Read as
	Category
=	equality	x = y means x and y represent the same thing or value.	1 + 1 = 2
	is equal to; equals
	everywhere
≠ <> !=	inequation	x ≠ y means that x and y do not represent the same thing or value. (The symbols != and <> are primarily from computer science. They are avoided in mathematical texts.)	1 ≠ 2
	is not equal to; does not equal
	everywhere
< > ≪ ≫	strict inequality	x < y means x is less than y. x > y means x is greater than y. x ≪ y means x is much less than y. x ≫ y means x is much greater than y.	3 <> 4. 0.003 ≪ 1000000
	is less than, is greater than, is much less than, is much greater than
	order theory
≤ ≥	inequality	x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y.	3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5
	is less than or equal to, is greater than or equal to
	order theory
∝	proportionality	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
	is proportional to
	everywhere
+	addition	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
	plus
	arithmetic
	disjoint union	A₁ + A₂ means the disjoint union of sets A₁ and A₂.	A₁ = {1, 2, 3, 4} ∧ A₂ = {2, 4, 5, 7} ⇒ A₁ + A₂ = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
	the disjoint union of ... and ...
	set theory
−	subtraction	9 − 4 means the subtraction of 4 from 9.	8 − 3 = 5
	minus
	arithmetic
	negative sign	−3 means the negative of the number 3.	−(−5) = 5
	negative ; minus
	arithmetic
	set-theoretic complement	A − B means the set that contains all the elements of A that are not in B.	{1,2,4} − {1,3,4} = {2}
	minus; without
	set theory
×	multiplication	3 × 4 means the multiplication of 3 by 4.	7 × 8 = 56
	times
	arithmetic
	Cartesian product	X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.	{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
	the Cartesian product of ... and ...; the direct product of ... and ...
	set theory
	cross product	u × v means the cross product of vectors u and v	(1,2,5) × (3,4,−1) = (−22, 16, − 2)
	cross
	vector algebra
·	multiplication	3 · 4 means the multiplication of 3 by 4.	7 · 8 = 56
	times
	arithmetic
	dot product	u · v means the dot product of vectors u and v	(1,2,5) · (3,4,−1) = 6
	dot
	vector algebra
÷ ⁄	division	6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.	2 ÷ 4 = .5 12 ⁄ 4 = 3
	divided by
	arithmetic
±	plus-minus	6 ± 3 means both 6 + 3 and 6 - 3.	The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
	plus or minus
	arithmetic
	plus-minus	10 ± 2 or eqivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.	If a = 100 ± 1 mm, then a is ≥ 99 mm and ≤ 101 mm.
	plus or minus
	measurment
∓	minus-plus	6 ± (3 ∓ 5) means both 6 + (3 - 5) and 6 - (3 + 5).	cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
	minus or plus
	arithmetic
√	square root	√x means the positive number whose square is x.	√4 = 2
	the principal square root of; square root
	real numbers
	complex square root	if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(i φ/2).	√(-1) = i
	the complex square root of … square root
	complex numbers
\|…\|	absolute value	\|x\| means the distance along the real line (or across the complex plane) between x and zero.	\|3\| = 3 \|–5\| = \|5\| \| i \| = 1 \| 3 + 4i \| = 5
	absolute value of
	numbers
	Euclidean distance	\|x – y\| means the Euclidean distance between x and y.	For x = (1,1), and y = (4,5), \|x – y\| = √([1–4]² + [1–5]²) = 5
	Euclidean distance between; Euclidean norm of
	Geometry
\|	divides	A single vertical bar is used to denote divisibility. a\|b means a divides b.	Since 15 = 3×5, it is true that 3\|15 and 5\|15.
	divides
	Number Theory
!	factorial	n ! is the product 1 × 2× ... × n.	4! = 1 × 2 × 3 × 4 = 24
	factorial
	combinatorics
T	transpose	Swap rows for columns	$A i j = (A T) j i$
	transpose
	matrix operations
~	probability distribution	X ~ D, means the random variable X has the probability distribution D.	X ~ N(0,1), the standard normal distribution
	has distribution
	statistics
⇒ → ⊃	material implication	A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. → may mean the same as ⇒, or it may have the meaning for functions given below. ⊃ may mean the same as ⇒, or it may have the meaning for superset given below.	x = 2 ⇒ x² = 4 is true, but x² = 4 ⇒ x = 2 is in general false (since x could be −2).
	implies; if … then
	propositional logic
⇔ ↔	material equivalence	A ⇔ B means A is true if B is true and A is false if B is false.	x + 5 = y +2 ⇔ x + 3 = y
	if and only if; iff
	propositional logic
¬ ˜	logical negation	The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front. (The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)	¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y)
	not
	propositional logic
∧	logical conjunction or meet in a lattice	The statement A ∧ B is true if A and B are both true; else it is false. For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).	n <>n >2 ⇔ n = 3 when n is a natural number.
	and; min
	propositional logic, lattice theory
∨	logical disjunction or join in a lattice	The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).	n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.
	or; max
	propositional logic, lattice theory
⊕ ⊻	exclusive or	The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.	(¬A) ⊕ A is always true, A ⊕ A is always false.
	xor
	propositional logic, Boolean algebra
	direct sum	The direct sum is a special way of combining several one modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).	Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)
	direct sum of
	Abstract algebra
∀	universal quantification	∀ x: P(x) means P(x) is true for all x.	∀ n ∈ ℕ: n² ≥ n.
	for all; for any; for each
	predicate logic
∃	existential quantification	∃ x: P(x) means there is at least one x such that P(x) is true.	∃ n ∈ ℕ: n is even.
	there exists
	predicate logic
∃!	uniqueness quantification	∃! x: P(x) means there is exactly one x such that P(x) is true.	∃! n ∈ ℕ: n + 5 = 2n.
	there exists exactly one
	predicate logic
:= ≡ :⇔	definition	x := y or x ≡ y means x is defined to be another name for y (Some writers use ≡ to mean congruence). P :⇔ Q means P is defined to be logically equivalent to Q.	cosh x := (1/2)(exp x + exp (−x)) A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
	is defined as
	everywhere
≅	congruence	△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
	is congruent to
	geometry
{ , }	set brackets	{a,b,c} means the set consisting of a, b, and c.	ℕ = { 1, 2, 3, …}
	the set of …
	set theory
{ : } { \| }	set builder notation	{x : P(x)} means the set of all x for which P(x) is true. {x \| P(x)} is the same as {x : P(x)}.	{n ∈ ℕ : n² <>
	the set of … such that
	set theory
∅ { }	empty set	∅ means the set with no elements. { } means the same.	{n ∈ ℕ : 1 < n² < class="Unicode">∅
	the empty set
	set theory
∈ ∉	set membership	a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S.	(1/2)⁻¹ ∈ ℕ 2⁻¹ ∉ ℕ
	is an element of; is not an element of
	everywhere, set theory
⊆ ⊂	subset	(subset) A ⊆ B means every element of A is also element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B. (Some writers use the symbol ⊂ as if it were the same as ⊆.)	(A ∩ B) ⊆ A ℕ ⊂ ℚ ℚ ⊂ ℝ
	is a subset of
	set theory
⊇ ⊃	superset	A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B. (Some writers use the symbol ⊃ as if it were the same as ⊇.)	(A ∪ B) ⊇ B ℝ ⊃ ℚ
	is a superset of
	set theory
∪	set-theoretic union	(exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both. "A or B, but not both." (inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B. "A or B or both".	A ⊆ B ⇔ (A ∪ B) = B (inclusive)
	the union of … and union
	set theory
∩	set-theoretic intersection	A ∩ B means the set that contains all those elements that A and B have in common.	{x ∈ ℝ : x² = 1} ∩ ℕ = {1}
	intersected with; intersect
	set theory
$Δ$	symmetric difference	$A Δ B$ means the set of elements in exactly one of A or B.	{1,5,6,8} $Δ$ {2,5,8} = {1,2,6}
	symmetric difference
	set theory
∖	set-theoretic complement	A ∖ B means the set that contains all those elements of A that are not in B.	{1,2,3,4} ∖ {3,4,5,6} = {1,2}
	minus; without
	set theory
( )	function application	f(x) means the value of the function f at the element x.	If f(x) := x², then f(3) = 3² = 9.
	of
	set theory
	precedence grouping	Perform the operations inside the parentheses first.	(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
	parentheses
	everywhere
f:X→Y	function arrow	f: X → Y means the function f maps the set X into the set Y.	Let f: ℤ → ℕ be defined by f(x) := x².
	from … to
	set theory
o	function composition	fog is the function, such that (fog)(x) = f(g(x)).	if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).
	composed with
	set theory
ℕ N	natural numbers	N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.	ℕ = {\|a\| : a ∈ ℤ, a ≠ 0}
	N
	numbers
ℤ Z	integers	ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ⁺ means {1, 2, 3, ...} = ℕ.	ℤ = {p, -p : p ∈ ℕ} ∪ {0}
	Z
	numbers
ℚ Q	rational numbers	ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.	3.14000... ∈ ℚ π ∉ ℚ
	Q
	numbers
ℝ R	real numbers	ℝ means the set of real numbers.	π ∈ ℝ √(−1) ∉ ℝ
	R
	numbers
ℂ C	complex numbers	ℂ means {a + b i : a,b ∈ ℝ}.	i = √(−1) ∈ ℂ
	C
	numbers
	arbitrary constant	C can be any number, most likely unknown; usually occurs when calculating antiderivatives.	if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C
	C
	integral calculus
∞	infinity	∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.	lim_x→0 1/\|x\| = ∞
	infinity
	numbers
π	pi	π is the ratio of a circle's circumference to its diameter. Its value is 3.14159265... .	A = π r² is the area of a circle with radius r π radians = 180° π ≈ 22 / 7
	pi
	Euclidean geometry
\|\|…\|\|	norm	\|\| x \|\| is the norm of the element x of a normed vector space.	\|\| x + y \|\| ≤ \|\| x \|\| + \|\| y \|\|
	norm of length of
	linear algebra
∑	summation	$\sum_{k=1}^{n}{a_k}$ means a₁ + a₂ + … + a_n.	$\sum_{k=1}^{4}{k^2}$ = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
	sum over … from … to … of
	arithmetic
∏	product	$\prod_{k=1}^na_k$ means a₁a₂···a_n.	$\prod_{k=1}^4(k+2)$ = (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360
	product over … from … to … of
	arithmetic
	Cartesian product	$\prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples (y₀, …, y_n).	$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$
	the Cartesian product of; the direct product of
	set theory
∐	coproduct
	coproduct over … from … to … of
	category theory
′	derivative	f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.	If f(x) := x², then f ′(x) = 2x
	… prime derivative of
	calculus
∫	indefinite integral or antiderivative	∫ f(x) dx means a function whose derivative is f.	∫x² dx = x³/3 + C
	indefinite integral of the antiderivative of
	calculus
	definite integral	∫_a^b f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.	∫₀^b x² dx = b³/3;
	integral from … to … of … with respect to
	calculus
∇	gradient	∇f (x₁, …, x_n) is the vector of partial derivatives (∂f / ∂x₁, …, ∂f / ∂x_n).	If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
	del, nabla, gradient of
	calculus
∂	partial derivative	With f (x₁, …, x_n), ∂f/∂x_i is the derivative of f with respect to x_i, with all other variables kept constant.	If f(x,y) := x²y, then ∂f/∂x = 2xy
	partial derivative of
	calculus
	boundary	∂M means the boundary of M	∂{x : \|\|x\|\| ≤ 2} = {x : \|\|x\|\| = 2}
	boundary of
	topology
⊥	perpendicular	x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.	If l ⊥ m and m ⊥ n then l \|\| n.
	is perpendicular to
	geometry
	bottom element	x = ⊥ means x is the smallest element.	∀x : x ∧ ⊥ = ⊥
	the bottom element
	lattice theory
\|\|	parallel	x \|\| y means x is parallel to y.	If l \|\| m and m ⊥ n then l ⊥ n.
	is parallel to
	geometry
⊧	entailment	A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true.	A ⊧ A ∨ ¬A
	entails
	model theory
⊢	inference	x ⊢ y means y is derived from x.	A → B ⊢ ¬B → ¬A
	infers or is derived from
	propositional logic, predicate logic
◅	normal subgroup	N ◅ G means that N is a normal subgroup of group G.	Z(G) ◅ G
	is a normal subgroup of
	group theory
/	quotient group	G/H means the quotient of group G modulo its subgroup H.	{0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
	mod
	group theory
	quotient set	A/~ means the set of all ~ equivalence classes in A.

	set theory
≈	isomorphism	G ≈ H means that group G is isomorphic to group H	Q / {1, −1} ≈ V, where Q is the quaternion group and V is the Klein four-group.
	is isomorphic to
	group theory
	approximately equal	x ≈ y means x is approximately equal to y	π ≈ 3.14159
	is approximately equal to
	everywhere
~	same order of magnitude	m ~ n, means the quantities m and n have the general size. (Note that ~ is used for an approximation that is poor, otherwise use ≈ .)	2 ~ 5 8 × 9 ~ 100 but π² ≈ 10
	roughly similar poorly approximates
	Approximation theory
<,>	inner product	<x,y> means the inner product between x and y, as defined in an inner product space.	The standard inner product between two vectors x = (2, 3) and y = (-1, 5) is: = 2×-1 + 3×5 = 13
	inner product of
	vector algebra
⊗	tensor product	V ⊗ U means the tensor product of V and U.	{1, 2, 3, 4} ⊗ {1,1,2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
	tensor product of
	linear algebra

# posted by wang @ 4:45 AM

Comments: Post a Comment

<< Home

biology

Thursday, November 16, 2006

Basic mathematical symbols

About Me

Links

archives