Simbol
|
Nama
|
Penjelasan
|
Contoh
|
Dibaca sebagai
|
|||
Kategori
|
|||
=
|
x = y berarti x and y
mewakili hal atau nilai yang sama.
|
1 + 1 = 2
|
|
sama dengan
|
|||
umum
|
|||
≠
|
x ≠ y berarti x dan y tidak
mewakili hal atau nilai yang sama.
|
1 ≠ 2
|
|
tidak sama dengan
|
|||
umum
|
|||
<
> |
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
|
|||
≤
≥ |
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
|
|||
+
|
4 + 6 berarti jumlah antara 4 dan 6.
|
2 + 7 = 9
|
|
tambah
|
|||
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 …
|
|||
−
|
9 − 4 berarti 9 dikurangi 4.
|
8 − 3 = 5
|
|
kurang
|
|||
−3 berarti negatif dari angka 3.
|
−(−5) = 5
|
||
negatif
|
|||
A − B berarti himpunan yang
mempunyai semua anggota dari A yang tidak terdapat pada B.
|
{1,2,4} − {1,3,4} = {2}
|
||
minus; without
|
|||
×
|
3 × 4 berarti perkalian 3 oleh 4.
|
7 × 8 = 56
|
|
kali
|
|||
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 …
|
|||
(1,2,5) × (3,4,−1) =
(−22, 16, − 2) |
|||
cross
|
|||
÷
/ |
6 ÷ 3 atau 6/3 berati 6 dibagi 3.
|
2 ÷ 4 = .5
12/4 = 3 |
|
bagi
|
|||
√
|
√x berarti bilangan positif yang kuadratnya x.
|
√4 = 2
|
|
akar kuadrat
|
|||
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
|
|||
| |
|
|3| = 3, |-5| = |5|
|i| = 1, |3+4i| = 5 |
||
nilai mutlak dari
|
|||
!
|
n! adalah hasil dari 1×2×...×n.
|
4! = 1 × 2 × 3 × 4 = 24
|
|
faktorial
|
|||
~
|
X ~ N(0,1), the standard normal distribution
|
||
has distribution
|
|||
⇒
→ ⊃ |
x = 2 ⇒ x2 = 4 is
true, but x2 = 4 ⇒ x = 2 is in general
false (since x could be −2).
|
||
implies; if .. then
|
|||
⇔
↔ |
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
|
|||
¬
˜ |
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
|
|||
∧
|
The statement A ∧ B
is true if A and B are both true; else it is false.
|
||
and
|
|||
∨
|
The statement A ∨ B
is true if A or B (or both) are true; if both are false, the
statement is false.
|
||
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
|
|||
∀
|
∀ x: P(x)
means P(x) is true for all x.
|
∀ n ∈ N:
n2 ≥ n.
|
|
for all; for any; for each
|
|||
∃
|
∃ x: P(x)
means there is at least one x such that P(x) is true.
|
∃ n ∈ N:
n is even.
|
|
there exists
|
|||
∃!
|
∃! x: P(x)
means there is exactly one x such that P(x) is true.
|
∃! n ∈ N:
n + 5 = 2n.
|
|
there exists exactly one
|
|||
:=
≡ :⇔ |
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 ...
|
|||
{ : }
{ | } |
{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 ...
|
|||
∅
{} |
∅ berarti himpunan yang tidak
memiliki elemen. {} juga berarti hal yang sama.
|
{n ∈ N : 1 < n2 <
4} = ∅
|
|
himpunan kosong
|
|||
∈
∉ |
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
|
|||
⊆
⊂ |
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
|
|||
⊇
⊃ |
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
|
|||
∪
|
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
|
|||
∩
|
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
|
|||
\
|
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
|
|||
( )
|
function application
|
f(x) berarti nilai fungsi f pada elemen
x.
|
Jika f(x) := x2,
maka f(3) = 32 = 9.
|
of
|
|||
precedence grouping
|
Perform the operations inside the parentheses first.
|
(8/4)/2 = 2/2 = 1, but 8/(4/2) =
8/2 = 4.
|
|
umum
|
|||
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
|
|||
o
|
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
|
|||
N
ℕ
|
N berarti {0,1,2,3,...}, but see the article on
natural numbers for a different convention.
|
{|a| : a ∈ Z} =
N
|
|
N
|
|||
Z
ℤ
|
Z berarti {...,−3,−2,−1,0,1,2,3,...}.
|
{a : |a| ∈ N} =
Z
|
|
Z
|
|||
Q
ℚ
|
Q berarti {p/q : p,q ∈ Z,
q ≠ 0}.
|
3.14 ∈ Q
π ∉ Q |
|
Q
|
|||
R
ℝ
|
R berarti {limn→∞ an :
∀ n ∈ N: an ∈ Q,
the limit exists}.
|
π ∈ R
√(−1) ∉ R |
|
R
|
|||
C
ℂ
|
C means {a + bi : a,b ∈ R}.
|
i = √(−1) ∈ C
|
|
C
|
|||
∞
|
∞ 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
|
|||
π
|
A = πr² adalah luas lingkaran dengan
jari-jari (radius) r
|
||
pi
|
|||
|| ||
|
||x+y|| ≤ ||x|| + ||y||
|
||
norm of; length of
|
|||
∑
|
∑k=1n ak
means a1 + a2 + ... + an.
|
∑k=14 k2 =
12 + 22 + 32 + 42 =
1 + 4 + 9 + 16 = 30
|
|
sum 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
|
|
product over ... from ... to ...
of
|
|||
∏n=13R
= Rn
|
|||
the Cartesian product of; the
direct product of
|
|||
'
|
If f(x) = x2,
then f '(x) = 2x
|
||
… prime; derivative of …
|
|||
∫
|
∫ f(x) dx means a
function whose derivative is f.
|
∫x2 dx = x3/3
+ C
|
|
indefinite integral of …; the antiderivative
of …
|
|||
∫0b x2
dx = b3/3;
|
|||
integral from ... to ... of ...
with respect to
|
|||
∇
|
∇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)
|
|
∂
|
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
|
|||
∂M means the boundary of M
|
∂{x : ||x|| ≤ 2} =
{x : || x || = 2} |
||
boundary of
|
|||
⊥
|
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
|
|||
x = ⊥ means x is the smallest element.
|
∀x : x ∧ ⊥ = ⊥
|
||
the bottom element
|
|||
|=
|
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
|
|||
|-
|
x ⊢ y means y is derived from x.
|
A → B ⊢ ¬B → ¬A
|
|
infers or is derived from
|
|||
◅
|
N ◅ G means that N is a
normal subgroup of group G.
|
Z(G) ◅ G
|
|
is a normal subgroup of
|
|||
/
|
{0, a, 2a, b, b+a,
b+2a} / {0, b} = {{0, b}, {a, b+a},
{2a, b+2a}}
|
||
mod
|
|||
≈
|
G ≈ H means that group G is isomorphic
to group H
|
||
is isomorphic to
|
|||
Kamis, 24 November 2011
Simbol Matematika
Langganan:
Posting Komentar (Atom)
Tidak ada komentar:
Posting Komentar