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Transcript
```Daftar simbol matematika
Simbol matematika dasar
Nama
Simbol
Dibaca sebagai
Penjelasan
Contoh
Kategori
kesamaan
=
sama dengan
x = y berarti x and y mewakili hal atau
nilai yang sama.
1+1=2
x ≠ y berarti x dan y tidak mewakili hal
atau nilai yang sama.
1≠2
umum
Ketidaksamaan
≠
tidak sama dengan
umum
<
ketidaksamaan
lebih kecil dari;
lebih besar dari
x < y berarti x lebih kecil dari y.
x > y means x lebih besar dari y.
>
order theory
≤
inequality
≥
x ≤ y berarti x lebih kecil dari atau sama
lebih kecil dari atau
dengan y.
sama dengan, lebih
besar dari atau sama
x ≥ y means x lebih besar dari atau sama
dengan
dengan y.
order theory
3<4
5>4
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
tambah
tambah
4 + 6 berarti jumlah antara 4 dan 6.
2+7=9
aritmatika
+
disjoint union
the disjoint union of A1 + A2 means the disjoint union of sets
… and …
A1 and A2.
teori himpunan
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)}
kurang
kurang
−
9 − 4 berarti 9 dikurangi 4.
8−3=5
−3 berarti negatif dari angka 3.
−(−5) = 5
aritmatika
tanda negatif
negatif
aritmatika
set-theoretic
complement
minus; without
A − B means the set that contains all the
elements of A that are not in B.
{1,2,4} − {1,3,4} = {2}
set theory
multiplication
kali
3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
aritmatika
Cartesian product
×
the Cartesian
product of … and
…; the direct
product of … and
…
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)}
teori himpunan
cross product
cross
u × v means the cross product of vectors (1,2,5) × (3,4,−1) =
u and v
(−22, 16, − 2)
vector algebra
÷
division
bagi
2 ÷ 4 = .5
6 ÷ 3 atau 6/3 berati 6 dibagi 3.
/
12/4 = 3
aritmatika
square root
√
√x berarti bilangan positif yang
√4 = 2
bilangan real
complex square root
if z = r exp(iφ) is represented in polar
the complex square
coordinates with -π < φ ≤ π, then √z = √r √(-1) = i
root of; square root
exp(iφ/2).
bilangan complex
absolute value
||
absolute value of
|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
n! is the product 1×2×...×n.
4! = 1 × 2 × 3 × 4 = 24
numbers
!
factorial
faktorial
combinatorics
~
probability
distribution
has distribution
X ~ D, means the random variable X has
the probability distribution D.
X ~ N(0,1), the standard
normal distribution
statistika
⇒
material implication A ⇒ B means if A is true then B is also
true; if A is false then nothing is said
implies; if .. then
→
→ may mean the same as ⇒, or it may
have the meaning for functions given
below.
propositional logic
⊃ may mean the same as ⇒, or it may
have the meaning for superset given
below.
⊃
⇔
material
equivalence
if and only if; iff
↔
¬
˜
A ⇔ B means A is true if B is true and A
is false if B is false.
logical negation
not
propositional logic
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.
and
logical disjunction
or join in a lattice
or
propositional logic,
lattice theory
exclusive or
xor
¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y)
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.
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.
propositional logic,
lattice theory
∨
x + 5 = y +2 ⇔ x + 3 = y
propositional logic
logical conjunction
or meet in a lattice
∧
x = 2 ⇒ x2 = 4 is true,
but x2 = 4 ⇒ x = 2 is in
general false (since x
could be −2).
The statement A ⊕ B is true when either (¬A) ⊕ A is always true,
A or B, but not both, are true. A ⊻ B
A ⊕ A is always false.
⊕
⊻
means the same.
propositional logic,
Boolean algebra
universal
quantification
∀
for all; for any; for
each
∀ x: P(x) means P(x) is true for all x.
∀ n ∈ N: n2 ≥ n.
∃ x: P(x) means there is at least one x
such that P(x) is true.
∃ n ∈ N: n is even.
∃! x: P(x) means there is exactly one x
such that P(x) is true.
∃! n ∈ N: n + 5 = 2n.
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).
cosh x := (1/2)(exp x +
exp (−x))
predicate logic
∃
existential
quantification
there exists
predicate logic
uniqueness
quantification
∃!
there exists exactly
one
predicate logic
:=
definition
is defined as
≡
everywhere
P :⇔ Q means P is defined to be
logically equivalent to Q.
A XOR B :⇔
(A ∨ B) ∧ ¬(A ∧ B)
:⇔
set brackets
{,}
the set of ...
{a,b,c} means the set consisting of a, b,
and c.
N = {0,1,2,...}
{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}
∅ berarti himpunan yang tidak memiliki
elemen. {} juga berarti hal yang sama.
{n ∈ N : 1 < n2 < 4} = ∅
teori himpunan
{:}
{|}
set builder notation
the set of ... such
that ...
teori himpunan
himpunan kosong
himpunan kosong
∅
teori himpunan
{}
∈
set membership
is an element of; is
not an element of
∉
everywhere, teori
himpunan
⊆
subset
is a subset of
⊂
teori himpunan
⊇
superset
⊃
is a superset of
teori himpunan
a ∈ S means a is an element of the set S;
a ∉ S means a is not an element of S.
A ⊆ B means every element of A is also
element of B.
(1/2)−1 ∈ N
2−1 ∉ N
A ∩ B ⊆ A; Q ⊂ R
A ⊂ B means A ⊆ B but A ≠ B.
A ⊇ B means every element of B is also
element of A.
A ∪ B ⊇ B; R ⊃ Q
A ⊃ B means A ⊇ B but A ≠ B.
set-theoretic union
∪
the union of ... and
...; 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
A ∩ B means the set that contains all
those elements that A and B have in
common.
{x ∈ R : x2 = 1} ∩ N =
{1}
teori himpunan
set-theoretic
intersection
∩
intersected with;
intersect
teori himpunan
\
set-theoretic
complement
minus; without
A \ B means the set that contains all those {1,2,3,4} \ {3,4,5,6} =
elements of A that are not in B.
{1,2}
teori himpunan
function application
()
of
teori himpunan
f(x) berarti nilai fungsi f pada elemen x.
Jika f(x) := x2, maka f(3) =
32 = 9.
precedence
grouping
Perform the operations inside the
parentheses first.
(8/4)/2 = 2/2 = 1, but
8/(4/2) = 8/2 = 4.
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.
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).
N means {0,1,2,3,...}, but see the article
on natural numbers for a different
convention.
{|a| : a ∈ Z} = N
Z means {...,−3,−2,−1,0,1,2,3,...}.
{a : |a| ∈ N} = Z
umum
arrow
f:X→ function
from ... to
Y
teori himpunan
function
composition
o
composed with
teori himpunan
natural numbers
N
N
ℕ
numbers
integers
Z
Z
ℤ
numbers
rational numbers
Q
Q
ℚ
numbers
3.14 ∈ Q
Q means {p/q : p,q ∈ Z, q ≠ 0}.
π∉Q
real numbers
R
R
ℝ
numbers
R means {limn→∞ an : ∀ n ∈ N: an ∈ Q,
the limit exists}.
π∈R
√(−1) ∉ R
complex numbers
C
C
ℂ
numbers
C means {a + bi : a,b ∈ R}.
i = √(−1) ∈ C
infinity
∞
infinity
numbers
∞ is an element of the extended number
line that is greater than all real numbers;
it often occurs in limits.
limx→0 1/|x| = ∞
π berarti perbandingan (rasio) antara
keliling lingkaran dengan diameternya.
lingkaran dengan jari-jari
||x|| is the norm of the element x of a
normed vector space.
||x+y|| ≤ ||x|| + ||y||
pi
π
pi
Euclidean geometry
norm
|| ||
norm of; length of
linear algebra
summation
∑
sum over ... from ...
∑k=1n ak means a1 + a2 + ... + an.
to ... of
∑k=14 k2 = 12 + 22 + 32 +
42 = 1 + 4 + 9 + 16 = 30
aritmatika
product
product over ...
from ... to ... of
∏
∏k=1n ak means a1a2···an.
∏k=14 (k + 2) = (1 +
2)(2 + 2)(3 + 2)(4 + 2) =
3 × 4 × 5 × 6 = 360
∏i=0nYi means the set of all (n+1)-tuples
(y0,...,yn).
∏n=13R = Rn
aritmatika
Cartesian product
the Cartesian
product of; the
direct product of
set theory
derivative
'
f '(x) is the derivative of the function f at
… prime; derivative
the point x, i.e., the slope of the tangent
of …
there.
kalkulus
If f(x) = x2, then f '(x) = 2x
indefinite integral
or antiderivative
∫
indefinite integral
∫ f(x) dx means a function whose
of …; the
derivative is f.
antiderivative of …
∫x2 dx = x3/3 + C
kalkulus
definite integral
∫ab f(x) dx means the signed area between
integral from ... to the x-axis and the graph of the function f ∫0b x2 dx = b3/3;
... of ... with respect between x = a and x = b.
to
kalkulus
∇
of
∇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)
kalkulus
partial derivative
∂
With f (x1, …, xn), ∂f/∂xi is the derivative
If f(x,y) = x2y, then ∂f/∂x
partial derivative of of f with respect to xi, with all other
= 2xy
variables kept constant.
kalkulus
boundary
boundary of
∂M means the boundary of M
∂{x : ||x|| ≤ 2} =
{x : || x || = 2}
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.
topology
perpendicular
is perpendicular to
⊥
geometri
bottom element
the bottom element x = ⊥ means x is the smallest element.
∀x : x ∧ ⊥ = ⊥
lattice theory
entailment
|=
entails
model theory
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
x ⊢ y means y is derived from x.
A → B ⊢ ¬B → ¬A
inference
|-
infers or is derived
from
propositional logic,
predicate logic
normal subgroup
◅
is a normal
subgroup of
N ◅ G means that N is a normal subgroup
Z(G) ◅ G
of group G.
group theory
quotient group
/
mod
group theory
≈
isomorphism
is isomorphic to
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}}
G ≈ H means that group G is isomorphic Q / {1, −1} ≈ V,
to group H
where Q is the quaternion
group theory
group and V is the Klein
four-group.
```