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Kamis, 17 November 2011

Simbol matematika dasar


Simbol matematika dasar
Simbol
Nama
Penjelasan
Contoh
Dibaca sebagai
Kategori
=
kesamaan
x = y berarti x and y mewakili hal atau nilai yang sama.
1 + 1 = 2
sama dengan
umum
Ketidaksamaan
xy berarti x dan y tidak mewakili hal atau nilai yang sama.
1 ≠ 2
tidak sama dengan
umum
<

>
ketidaksamaan
x < y berarti x lebih kecil dari y.

x > y means x lebih besar dari y.
3 < 4
5 > 4
lebih kecil dari; lebih besar dari
order theory


inequality
x ≤ y berarti x lebih kecil dari atau sama dengan y.

x ≥ y berarti x lebih besar dari atau sama dengan y.
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
lebih kecil dari atau sama dengan, lebih besar dari atau sama dengan
order theory
+
tambah
4 + 6 berarti jumlah antara 4 dan 6.
2 + 7 = 9
tambah
aritmatika
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 …
teori himpunan
kurang
9 − 4 berarti 9 dikurangi 4.
8 − 3 = 5
kurang
aritmatika
tanda negatif
−3 berarti negatif dari angka 3.
−(−5) = 5
negatif
aritmatika
set-theoretic complement
A − B berarti himpunan yang mempunyai semua anggota dari A yang tidak terdapat pada B.
{1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
multiplication
3 × 4 berarti perkalian 3 oleh 4.
7 × 8 = 56
kali
aritmatika
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 …
teori himpunan
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
÷

/
division
6 ÷ 3 atau 6/3 berati 6 dibagi 3.
2 ÷ 4 = .5

12/4 = 3
bagi
aritmatika
square root
x berarti bilangan positif yang kuadratnya x.
√4 = 2
akar kuadrat
bilangan real
complex square root
if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(iφ/2).
√(-1) = i
the complex square root of; square root
Bilangan kompleks
| |
absolute value
|x| means the distance in the real line (or the complex plane) between x and zero.
|3| = 3, |-5| = |5|
|i| = 1, |3+4i| = 5
nilai mutlak dari
numbers
!
factorial
n! adalah hasil dari 1×2×...×n.
4! = 1 × 2 × 3 × 4 = 24
faktorial
combinatorics
~
probability distribution
X ~ D, means the random variable X has the probability distribution D.
X ~ N(0,1), the standard normal distribution
has distribution
statistika




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.
¬(¬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.
n < 4    n >2    n = 3 when n is a natural number.
and
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.
n ≥ 4    n ≤ 2   n ≠ 3 when n is a natural number.
or
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
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 (but note that ≡ can also mean other things, such as 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
{ , }
set brackets
{a,b,c} means the set consisting of a, b, and c.
N = {0,1,2,...}
the set of ...
teori himpunan
{ : }

{ | }
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 < 20} = {0,1,2,3,4}
the set of ... such that ...
teori himpunan



{}
himpunan kosong
berarti himpunan yang tidak memiliki elemen. {} juga berarti hal yang sama.
{n  N : 1 < n2 < 4} =
himpunan kosong
teori himpunan


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, teori himpunan


subset
A  B means every element of A is also element of B.

A 
B means A  B but A ≠ B.
A ∩ B A; Q  R
is a subset of
teori himpunan


superset
A  B means every element of B is also element of A.

A 
B means A  B but A ≠ B.
A  B B; R  Q
is a superset of
teori himpunan
set-theoretic union
A  B means the set that contains all the elements from A and also all those from B, but no others.
A  B    A  B = B
the union of ... and ...; union
teori himpunan
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
teori himpunan
\
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
teori himpunan
( )
function application
f(x) berarti nilai fungsi f pada elemen x.
Jika f(x) := x2, maka f(3) = 32 = 9.
of
teori himpunan
precedence grouping
Perform the operations inside the parentheses first.
(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.

umum
f:XY
function arrow
fX → Y means the function f maps the set X into the set Y.
Let fZ → N be defined by f(x) = x2.
from ... to
teori himpunan
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
teori himpunan

N

Bilangan asli
N berarti {0,1,2,3,...}, but see the article on natural numbers for a different convention.
{|a| : a  Z} = N
N
Bilangan

Z

Bilangan bulat
Z berarti {...,−3,−2,−1,0,1,2,3,...}.
{a : |a N} = Z
Z
Bilangan

Q

Bilangan rasional
Q berarti {p/q : p,q  Z, q ≠ 0}.
3.14  Q

π 
Q
Q
Bilangan

R

Bilangan real
R berarti {limn→∞ an :  n  N: an  Q, the limit exists}.
π  R

√(−1) 
 R
R
Bilangan

C

Bilangan kompleks
C means {a + bi : a,b  R}.
i = √(−1)  C
C
Bilangan
infinity
∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.
limx→0 1/|x| = ∞
infinity
numbers
π
pi
π berarti perbandingan (rasio) antara keliling lingkaran dengan diameternya.
A = πr² adalah luas lingkaran dengan jari-jari (radius) r
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
k=1n ak means a1 + a2 + ... + an.
k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
sum over ... from ... to ... of
aritmatika
product
k=1n ak means a1a2···an.
k=14 (k + 2) = (1  + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
product over ... from ... to ... of
aritmatika
Cartesian product
i=0nYi means the set of all (n+1)-tuples (y0,...,yn).
n=13R = Rn
the Cartesian product of; the direct product of
set theory
'
derivative
f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there.
If f(x) = x2, then f '(x) = 2x
… prime; derivative of …
kalkulus
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 …
kalkulus
definite integral
ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.
∫0b x2  dx = b3/3;
integral from ... to ... of ... with respect to
kalkulus
gradient
f (x1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn).
If f (x,y,z) = 3xy + z² then f = (3y, 3x, 2z)
del, nabla, gradient of
kalkulus
partial derivative
With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant.
If f(x,y) = x2y, then ∂f/∂x = 2xy
partial derivative of
kalkulus
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 lm and mn then l || n.
is perpendicular to
geometri
bottom element
x = means x is the smallest element.
x : x =
the bottom element
lattice theory
|=
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.
AB ¬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
isomorphism
GH 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

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