Simbol matematika dasar
Simbol
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Nama
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Penjelasan
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Contoh
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Dibaca sebagai
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Kategori
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=
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x = y berarti x and y
mewakili hal atau nilai yang sama.
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1 + 1 = 2
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sama dengan
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umum
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≠
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x ≠ y berarti x dan y tidak
mewakili hal atau nilai yang sama.
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1 ≠ 2
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tidak sama dengan
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umum
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<
> |
x < y berarti x lebih kecil
dari y.
x > y means x lebih besar dari y. |
3 < 4
5 > 4 |
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lebih kecil dari; lebih besar dari
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≤
≥ |
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 |
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lebih kecil dari atau sama dengan,
lebih besar dari atau sama dengan
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+
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4 + 6 berarti jumlah antara 4 dan 6.
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2 + 7 = 9
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tambah
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A1 + A2 means the disjoint union of
sets A1 and A2.
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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)} |
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the disjoint union of … and …
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−
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9 − 4 berarti 9 dikurangi 4.
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8 − 3 = 5
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kurang
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−3 berarti negatif dari angka 3.
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−(−5) = 5
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negatif
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A − B berarti himpunan yang
mempunyai semua anggota dari A yang tidak terdapat pada B.
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{1,2,4} − {1,3,4} = {2}
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minus; without
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×
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3 × 4 berarti perkalian 3 oleh 4.
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7 × 8 = 56
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kali
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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.
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{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
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the Cartesian product of … and …;
the direct product of … and …
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(1,2,5) × (3,4,−1) =
(−22, 16, − 2) |
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cross
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÷
/ |
6 ÷ 3 atau 6/3 berati 6 dibagi 3.
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2 ÷ 4 = .5
12/4 = 3 |
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bagi
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√
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√x berarti bilangan positif yang kuadratnya x.
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√4 = 2
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akar kuadrat
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if z = r exp(iφ) is represented
in polar coordinates with -π < φ ≤ π, then √z = √r exp(iφ/2).
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√(-1) = i
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the complex square root of; square
root
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|3| = 3, |-5| = |5|
|i| = 1, |3+4i| = 5 |
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nilai mutlak dari
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!
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n! adalah hasil dari 1×2×...×n.
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4! = 1 × 2 × 3 × 4 = 24
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faktorial
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~
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X ~ N(0,1), the standard normal distribution
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has distribution
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⇒
→ ⊃ |
x = 2 ⇒ x2 = 4 is
true, but x2 = 4 ⇒ x = 2 is in general
false (since x could be −2).
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implies; if .. then
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⇔
↔ |
A ⇔ B means A is true
if B is true and A is false if B is false.
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x + 5 = y +2 ⇔ x +
3 = y
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if and only if; iff
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¬
˜ |
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) |
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not
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∧
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The statement A ∧ B
is true if A and B are both true; else it is false.
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and
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∨
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The statement A ∨ B
is true if A or B (or both) are true; if both are false, the
statement is false.
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or
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⊕
⊻
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The statement A ⊕ B
is true when either A or B, but not both, are true. A ⊻ B
means the same.
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(¬A) ⊕ A is always true, A
⊕ A is always false.
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xor
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∀
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∀ x: P(x)
means P(x) is true for all x.
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∀ n ∈ N:
n2 ≥ n.
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for all; for any; for each
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∃
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∃ x: P(x)
means there is at least one x such that P(x) is true.
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∃ n ∈ N:
n is even.
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there exists
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∃!
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∃! x: P(x)
means there is exactly one x such that P(x) is true.
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∃! n ∈ N:
n + 5 = 2n.
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there exists exactly one
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:=
≡ :⇔ |
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) |
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is defined as
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everywhere
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{ , }
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set brackets
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{a,b,c} means the set
consisting of a, b, and c.
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N = {0,1,2,...}
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the set of ...
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{ : }
{ | } |
{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)}.
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{n ∈ N : n2 < 20} =
{0,1,2,3,4}
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the set of ... such that ...
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∅
{} |
∅ berarti himpunan yang tidak
memiliki elemen. {} juga berarti hal yang sama.
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{n ∈ N : 1 < n2 <
4} = ∅
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himpunan kosong
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∈
∉ |
set membership
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a ∈ S means a is an
element of the set S; a ∉ S means a is not an
element of S.
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(1/2)−1 ∈ N
2−1 ∉ N |
is an element of; is not an
element of
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everywhere, teori himpunan
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⊆
⊂ |
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
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is a subset of
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⊇
⊃ |
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
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is a superset of
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∪
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A ∪ B means the set that
contains all the elements from A and also all those from B, but
no others.
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A ⊆ B ⇔ A ∪ B =
B
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the union of ... and ...; union
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∩
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A ∩ B means the set that contains all
those elements that A and B have in common.
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{x ∈ R : x2 =
1} ∩ N = {1}
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intersected with; intersect
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\
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A \ B means the set that contains all
those elements of A that are not in B.
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{1,2,3,4} \ {3,4,5,6} = {1,2}
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minus; without
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( )
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function application
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f(x) berarti nilai fungsi f pada elemen
x.
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Jika f(x) := x2,
maka f(3) = 32 = 9.
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of
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precedence grouping
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Perform the operations inside the parentheses first.
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(8/4)/2 = 2/2 = 1, but 8/(4/2) =
8/2 = 4.
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umum
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f:X→Y
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function arrow
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f: X → Y means the function f
maps the set X into the set Y.
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Let f: Z → N be
defined by f(x) = x2.
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from ... to
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o
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fog is the function, such that (fog)(x)
= f(g(x)).
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if f(x) = 2x, and g(x)
= x + 3, then (fog)(x) = 2(x + 3).
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composed with
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N
ℕ
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N berarti {0,1,2,3,...}, but see the article on
natural numbers for a different convention.
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{|a| : a ∈ Z} =
N
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N
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Z
ℤ
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Z berarti {...,−3,−2,−1,0,1,2,3,...}.
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{a : |a| ∈ N} =
Z
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Z
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Q
ℚ
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Q berarti {p/q : p,q ∈ Z,
q ≠ 0}.
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3.14 ∈ Q
π ∉ Q |
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Q
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R
ℝ
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R berarti {limn→∞ an :
∀ n ∈ N: an ∈ Q,
the limit exists}.
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π ∈ R
√(−1) ∉ R |
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R
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C
ℂ
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C means {a + bi : a,b ∈ R}.
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i = √(−1) ∈ C
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C
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∞
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∞ is an element of the extended number line that is greater than all real
numbers; it often occurs in limits.
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limx→0 1/|x| = ∞
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infinity
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π
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A = πr² adalah luas lingkaran dengan
jari-jari (radius) r
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pi
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||x+y|| ≤ ||x|| + ||y||
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norm of; length of
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∑
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∑k=1n ak
means a1 + a2 + ... + an.
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∑k=14 k2 =
12 + 22 + 32 + 42 =
1 + 4 + 9 + 16 = 30
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sum over ... from ... to ... of
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∏
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∏k=1n ak
means a1a2···an.
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∏k=14 (k +
2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 ×
4 × 5 × 6 = 360
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product over ... from ... to ...
of
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∏n=13R
= Rn
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the Cartesian product of; the
direct product of
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'
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If f(x) = x2,
then f '(x) = 2x
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… prime; derivative of …
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∫
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∫ f(x) dx means a
function whose derivative is f.
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∫x2 dx = x3/3
+ C
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indefinite integral of …; the antiderivative
of …
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∫0b x2
dx = b3/3;
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integral from ... to ... of ...
with respect to
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∇
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∇f (x1, …, xn)
is the vector of partial derivatives (df / dx1, …, df
/ dxn).
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If f (x,y,z) = 3xy
+ z² then ∇f = (3y, 3x, 2z)
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∂
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With f (x1, …, xn),
∂f/∂xi is the derivative of f with respect to xi,
with all other variables kept constant.
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If f(x,y) = x2y, then ∂f/∂x
= 2xy
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partial derivative of
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∂M means the boundary of M
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∂{x : ||x|| ≤ 2} =
{x : || x || = 2} |
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boundary of
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⊥
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x ⊥ y means x is perpendicular to y;
or more generally x is orthogonal to y.
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If l⊥m and m⊥n then l || n.
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is perpendicular to
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x = ⊥ means x is the smallest element.
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∀x : x ∧ ⊥ = ⊥
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the bottom element
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A ⊧ B means the sentence A entails the
sentence B, that is every model in which A is true, B is also true.
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A ⊧ A ∨ ¬A
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entails
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x ⊢ y means y is derived from x.
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A → B ⊢ ¬B → ¬A
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infers or is derived from
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◅
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N ◅ G means that N is a
normal subgroup of group G.
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Z(G) ◅ G
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is a normal subgroup of
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/
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{0, a, 2a, b, b+a,
b+2a} / {0, b} = {{0, b}, {a, b+a},
{2a, b+2a}}
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mod
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≈
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G ≈ H means that group G is isomorphic
to group H
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is isomorphic to
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