গাণিতিক চিহ্নের সারণি - Wikipedia

# গাণিতিক চিহ্নের সারণি

### উইকিপিডিয়া, মুক্ত বিশ্বকোষ থেকে

## [সম্পাদনা করুন] মূল চিহ্নসমূহের তালিকা

চিহ্ন | নাম | ব্যাখ্যা | উদাহরণ
--- | --- | --- | ---
যেভাবে পড়তে হবে
বিষয়শ্রেণী
= | সমতা | x = y অর্থ x ও y একই জিনিস অথবা মান | ১ + ২ = ৩
সমান সমান; সমান
সবক্ষেত্রে
≠ <> != | অসমান | x ≠ y অর্থ x এবং y একই জিনিস নয় অথবা তাদের মান সমান নয় | 1 ≠ 2
অসমান
সবক্ষেত্রে
< > ≪ ≫ | অসমতা | x < y অর্থ x y থেকে ক্ষুদ্রতর x > y অর্থ x y থেকে বৃহত্তর x ≪ y অর্থ x y থেকে অনেক ক্ষুদ্রতর x ≫ y অর্থ x y থেকে অনেক বৃহত্তর | ৩ < ৪ ৫ > ৩ ০.০০৭ ≪ ১০০০০০০০
ক্ষুদ্রতর, বৃহত্তর, অনেক ছোট, অনেক বড়
order theory
≤ ≥ | অসমতা | x ≤ y অর্থ x y এর চেয়ে ছোট বা সমান x ≥ y অর্থ x y এর চেয়ে বড় বা সমান | ৩ ≤ ৪ এবং ৫ ≤ ৫ ৫ ≥ ৪ এবং ৫ ≥ ৫
ক্ষুদ্রতর অথবা সমান, বৃহত্তর অথবা সমান
order theory
∝ | সমানুপাত | y ∝ x অর্থ y = kx , কোন ধ্রুবক k এর জন্য | যদি y = 2x, তাহলে y ∝ x
সমানুপাতিক
সবক্ষেত্রে
+ | যোগ | ৪ + ৬ অর্থ ৪ ও ৬ এর যোগফল | ২ + ৭ = ৯
যোগ
গণিত
disjoint union | A1 + A2 অর্থ 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 অর্থ the subtraction of 4 from 9. | 8 − 3 = 5
minus
arithmetic
negative sign | −3 অর্থ the negative of the number 3. | −(−5) = 5
negative ; minus
arithmetic
set-theoretic complement | A − B অর্থ 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 অর্থ the multiplication of 3 by 4. | 7 × 8 = 56
times
arithmetic
Cartesian product | X×Y অর্থ 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 অর্থ the cross product of vectors u and v | (1,2,5) × (3,4,−1) = (−22, 16, − 2)
cross
vector algebra
÷ / | division | 6 ÷ 3 or 6/3 অর্থ the division of 6 by 3. | 2 ÷ 4 = .5 12/4 = 3
divided by
arithmetic
√ | square root | √x অর্থ 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(iφ/2). | √(-1) = i
the complex square root of; square root
complex numbers
| | | absolute value | |x| অর্থ the distance in the real line (or the complex plane) between x and zero. | |3| = 3, |-5| = |5| |i| = 1, |3+4i| = 5
absolute value of
numbers
! | factorial | n! is the product 1 × 2× ... × n. | 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
~ | probability distribution | X ~ D, অর্থ the random variable X has the probability distribution D. | X ~ N(0,1), the standard normal distribution
has distribution
statistics
⇒ → ⊃ | material implication | A ⇒ B অর্থ 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 অর্থ 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 অর্থ the same. | (¬A) ⊕ A is always true, A ⊕ A is always false.
xor
propositional logic, Boolean algebra
∀ | universal quantification | ∀ x: P(x) অর্থ P(x) is true for all x. | ∀ n ∈ N: n2 ≥ n.
for all; for any; for each
predicate logic
∃ | existential quantification | ∃ x: P(x) অর্থ 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) অর্থ 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 অর্থ x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence). P :⇔ Q অর্থ 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
সবক্ষেত্রে
{ , } | set brackets | {a,b,c} অর্থ the set consisting of a, b, and c. | N = {0, 1, 2, ...}
the set of ...
set theory
{ : } { | } | set builder notation | {x : P(x)} অর্থ 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 ...
set theory
∅ {} | empty set | ∅ অর্থ the set with no elements. {} অর্থ the same. | {n ∈ N : 1 < n2 < 4} = ∅
the empty set
set theory
∈ ∉ | set membership | a ∈ S অর্থ a is an element of the set S; a ∉ S অর্থ a is not an element of S. | (1/2)−1 ∈ N 2−1 ∉ N
is an element of; is not an element of
সবক্ষেত্রে, set theory
⊆ ⊂ | subset | (subset) A ⊆ B অর্থ every element of A is also element of B. (proper subset) A ⊂ B অর্থ A ⊆ B but A ≠ B. | A ∩ B ⊆ A; Q ⊂ R
is a subset of
set theory
⊇ ⊃ | superset | A ⊇ B অর্থ every element of B is also element of A. A ⊃ B অর্থ A ⊇ B but A ≠ B. | A ∪ B ⊇ B; R ⊃ Q
is a superset of
set theory

∪

set-theoretic union

(exclusive) A ∪ B অর্থ 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 অর্থ 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 অর্থ 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
\ | set-theoretic complement | A \ B অর্থ 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) অর্থ 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
সবক্ষেত্রে
f:X→Y | function arrow | f: X → Y অর্থ 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 অর্থ {0, 1, 2, 3, ...}, but see the article on natural numbers for a different convention. | {|a| : a ∈ Z} = N
N
numbers
Z ℤ | integers | Z অর্থ {..., −3, −2, −1, 0, 1, 2, 3, ...}. | {a, -a : a ∈ N} = Z
Z
numbers
Q ℚ | rational numbers | Q অর্থ {p/q : p,q ∈ Z, q ≠ 0}. | 3.14 ∈ Q π ∉ Q
Q
numbers
R ℝ | real numbers | R অর্থ the set of real numbers. | π ∈ R √(−1) ∉ R
R
numbers
C ℂ | complex numbers | C অর্থ {a + bi : 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. | limx→0 1/|x| = ∞
infinity
numbers
π | pi | π is the ratio of a circle's circumference to its diameter. Its value is 3.1415.... | A = πr² is the area of a circle with 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

অর্থ a1 + a2 + ... + an.

= 12 + 22 + 32 + 42

    = 1 + 4 + 9 + 16 = 30

sum over ... from ... to ... of
arithmetic

∏

product

অর্থ a1a2•••an.

= (1+2)(2+2)(3+2)(4+2)

    = 3 × 4 × 5 × 6 = 360

product over ... from ... to ... of
arithmetic

Cartesian product

অর্থ the set of all (n+1)-tuples

    (y0,...,yn).

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 to f at x. | If f(x) := x2, then f '(x) = 2x
... prime; derivative of ...
calculus
∫ | indefinite integral or antiderivative | ∫ f(x) dx অর্থ a function whose derivative is f. | ∫x2 dx = x3/3 + C
indefinite integral of ...;; the antiderivative of ...
calculus
definite integral | ∫ab f(x) dx অর্থ 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
calculus
∇ | 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
calculus
∂ | 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
calculus
boundary | ∂M অর্থ the boundary of M | ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
⊥ | perpendicular | x ⊥ y অর্থ 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 = ⊥ অর্থ x is the smallest element. | ∀x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
⊧ | entailment | A ⊧ B অর্থ 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 অর্থ y is derived from x. | A → B ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
◅ | normal subgroup | N ◅ G অর্থ that N is a normal subgroup of group G. | Z(G) ◅ G
is a normal subgroup of
group theory
/ | quotient group | G/H অর্থ 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/~ অর্থ the set of all ~ equivalence classes in A.
set theory
≈ | isomorphism | G ≈ H অর্থ 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 অর্থ x is approximately equal to y | π ≈ 3.14159
is approximately equal to
সবক্ষেত্রে
⊗ | tensor product | V ⊗ U অর্থ 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

## [সম্পাদনা করুন] আরও দেখুন

- Mathematical alphanumeric symbols
- গাঠনিক ধ্রুবক
- পদার্থবিজ্ঞানের চলক
- আই এস ও ৩১-১১

## [সম্পাদনা করুন] বহিঃসংযোগ

- Jeff Miller: প্রাচীনকালে ব্যবহৃত গাণিতিক চিহ্নসমূহ
- TCAEP - পদার্থবিজ্ঞান ইনস্টিটিউশন
- গাণিতিক চিহ্নসমূহের GIF এবং PNG চিত্র

## [সম্পাদনা করুন] বিশেষ চিহ্নসমূহ

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