# கணிதக் குறியீடுகள்

தாவிச் செல்லவும்: வழிசெலுத்தல், தேடல்

கணித தகவல்களை வெளிப்படுத்த கணிதக் குறியீடுகள் பயன்படுகின்றன. எழுத்துக்கள் எப்படி மொழியூடாக தகவல்களை வெளிப்படுத்த அவசியமோ அதேபோல் குறியீடுகள் கணிதத்தினூடாக தகவல்களை வெளிப்படுத்த அவசியம். ஒரு மொழியை அறிய, பயன்படுத்த எப்படி அதன் எழுத்துக்களை அறிவது அவசியமோ அதேபோல் கணிதத்தை அறிய, பயன்படுத்த கணிதக் குறியீடுகளை அறிவது அவசியம். எண்கள், செயற்பாட்டுக் குறியீடுகள், கருத்துருக் குறியீடுகள், சமன்பாடுகள் என பலநிலையிலான குறியீடுகள் கணிதத்தில் உண்டு.

## அடிப்படை கணிதக் குறியீடுகள் அட்டவணை

குறியீடு பெயர் விளக்கம் எடுத்துக்காட்டு
பலுக்கும் முறை
பகுப்பு
=
சமம் காட்டாக 2 + 3 = 5 என்பது ஒரு சமன்பாடு. இதனை 2 கூட்டல் 3 ஈடு 5 என்று படிக்கலாம், அல்லது 2 கூட்டல் 3 சமம் 5 என்று படிக்கலாம். அதே போல 2 + 4 = 3 x 2 என்பதும் ஒரு சமன்பாடு. 1 + 1 = 2
சமமாக, ஈடாக
எங்கும்

<>

!=
சமனிலி x ≠ y என்பது x ம் y யும் ஒன்றல்ல, ஒரே மதிப்பைக் கொள்ளவில்லை. .

(குறியீடுகள் != ம் <> கணினியியலில் பயன்படுகிறது.)
1 ≠ 2
சமமில்லை
<

>

strict inequality x < y என்பது x ஐவிடச் சிறியது y.

x > y என்பது x yயிலும் பெரியது.

x ≪ y என்பது x y ஐவிட மிகச் சிறியது.

x ≫ y என்பது x yஐவிடப் மிகவும் பெரியது.
3 < 4
5 > 4.

0.003 ≪ 1000000

is less than, is greater than, is much less than, is much greater than
order theory

<=

>=
inequality x ≤ y means x is less than or equal to y.

x ≥ y means x is greater than or equal to y.

(The symbols <= and >= are primarily from computer science. They are avoided in mathematical texts.)
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is less than or equal to, is greater than or equal to
order theory
proportionality yx means that y = kx for some constant k. if y = 2x, then yx
is proportional to; varies as
everywhere
+
கூட்டல் 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
எண்கணிதம்
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 …
set theory
கழித்தல் 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5
minus
எண்கணிதம்
negative sign −3 means the negative of the number 3. −(−5) = 5
negative; minus
எண்கணிதம்
set-theoretic complement A − B means the set that contains all the elements of A that are not in B.

∖ can also be used for set-theoretic complement as described below.
{1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
பெருக்கல் 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
times
எண்கணிதம்
கார்ட்டீசியன் பெருக்கற்பலன் X×Y means the set of all வரிசைச் சோடி 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
குறுக்குப் பெருக்கு u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross
vector algebra
·
பெருக்கல் 3 · 4 means the multiplication of 3 by 4. 7 · 8 = 56
times
எண்கணிதம்
புள்ளிப் பெருக்கல் u · v means the dot product of vectors u and v (1,2,5) · (3,4,−1) = 6
dot
vector algebra
÷

division 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = .5

12 ⁄ 4 = 3
divided by
எண்கணிதம்
±
plus-minus 6 ± 3 means both 6 + 3 and 6 – 3. The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
plus or minus
எண்கணிதம்
plus-minus 10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
plus or minus
அளவியல்
minus-plus 6 ± (3 5) means both 6 + (3 – 5) and 6 – (3 + 5). cos(x ± y) = cos(x) cos(y) sin(x) sin(y).
minus or plus
எண்கணிதம்
வர்க்கமூலம் x means the positive number whose square is x. √4 = 2
the principal square root of; square root
மெய்யெண்
complex square root if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(i φ/2). √(-1) = i
the complex square root of …

square root
சிக்கலெண்
|…|
தனி மதிப்பு or modulus |x| means the distance along the real line (or across the complex plane) between x and zero. |3| = 3

|–5| = |5|

i | = 1

| 3 + 4i | = 5
absolute value (modulus) of
எண்s
யூக்ளிடிய தொலைவு |x – y| means the Euclidean distance between x and y. For x = (1,1), and y = (4,5),
|x – y| = √([1–4]2 + [1–5]2) = 5
Euclidean distance between; Euclidean norm of
வடிவவியல்
அணிக்கோவை |A| means the determinant of the matrix A $\begin{vmatrix} 1&2 \\ 2&4 \\ \end{vmatrix} = 0$
determinant of
அணி (கணிதம்)
|
divides A single vertical bar is used to denote divisibility.
a|b means a divides b.
Since 15 = 3×5, it is true that 3|15 and 5|15.
divides
Number Theory
!
தொடர் பெருக்கம் n ! is the product 1 × 2× … × n. 4! = 1 × 2 × 3 × 4 = 24
factorial
சேர்வியல் (கணிதம்)
T
transpose Swap rows for columns $A_{ij} = (A^T)_{ji}$
transpose
அணி (கணிதம்)s
~
நிகழ்தகவுப் பரவல் X ~ D, means the சமவாய்ப்பு மாறி X has the probability distribution D. X ~ N(0,1), the இயல்நிலைப் பரவல்
has distribution
புள்ளியியல்
Row equivalence A~B means that B can be generated by using a series of elementary row operations on A $\begin{bmatrix} 1&2 \\ 2&4 \\ \end{bmatrix} \sim \begin{bmatrix} 1&2 \\ 0&0 \\ \end{bmatrix}$
is row equivalent to
அணி (கணிதம்)

material implication AB 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, Heyting algebra

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.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred.)
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
logical conjunction or meet in a lattice The statement AB is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) ∧ B(x) is used to mean min(A(x), B(x)).
n < 4  ∧  n >2  ⇔  n = 3 when n is a இயல் எண்.
and; min
propositional logic, lattice theory
logical disjunction or join in a lattice The statement AB is true if A or B (or both) are true; if both are false, the statement is false.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).
n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a இயல் எண்.
or; max
propositional logic, lattice theory

exclusive or The statement AB is true when either A or B, but not both, are true. A B means the same. A) ⊕ A is always true, AA is always false.
xor
propositional logic, Boolean algebra
direct sum The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, is only for logic).

Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = )
direct sum of
Abstract algebra
universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ 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 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 + 5 = 2n.
there exists exactly one
predicate logic
:=

:⇔
வரைவிலக்கணம் x := y or x ≡ y means x is defined to be another name for y

(Some writers useto mean 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
congruence △ABC △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
is congruent to
வடிவவியல்
congruence relation a ≡ b (mod n) means a − b is divisible by n 5 ≡ 11 (mod 3)
… is congruent to … modulo …
சமானம், மாடுலோ n
{ , }
தொடை brackets {a,b,c} means the set consisting of a, b, and c.  = { 1, 2, 3, …}
the set of …
set theory
{ : }

{ | }
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 ∈  : n2 < 20} = { 1, 2, 3, 4}
the set of … such that
set theory

{ }
சூனியத்தொடை means the set with no elements. { } means the same. {n ∈  : 1 < n2 < 4} =
the empty set
set theory

set membership a ∈ S என்பது a , Sதொடையின் மூலகமாகும் ; a  S என்பது a ,Sதொடையின் மூலகமல்ல என்றும் குறித்து நிற்கும் . (1/2)−1 ∈

2−1
மூலகம் ; மூலகமன்று
everywhere, set theory

உபதொடை (subset) A ⊆ B means every element of A is also element of B.

(proper subset) A ⊂ B means A ⊆ B but A ≠ B.

(Some writers use the symbol ⊂ as if it were the same as ⊆.)
(A ∩ B) ⊆ A

⊂

⊂
is a subset of
set theory

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

A ⊃ B means A ⊇ B but A ≠ B.

(Some writers use the symbol ⊃ as if it were the same as ⊇.)
(A ∪ B) ⊇ B

⊃
is a superset of
set theory
set-theoretic union (exclusive) A ∪ B means 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 means 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 means the set that contains all those elements that A and B have in common. {x ∈  : x2 = 1} ∩  = {1}
intersected with; intersect
set theory
$\Delta$
symmetric difference $A\Delta B$ means the set of elements in exactly one of A or B. {1,5,6,8} $\Delta$ {2,5,8} = {1,2,6}
symmetric difference
set theory
set-theoretic complement A  B means the set that contains all those elements of A that are not in B.

− can also be used for set-theoretic complement as described above.
{1,2,3,4}  {3,4,5,6} = {1,2}
minus; without
set theory
( )
function application f(x) means 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
everywhere
f:XY
function arrow fX → Y means the function f maps the set X into the set Y. Let f →  be defined by f(x) := x2.
from … to
set theory,type theory
o
சார்புகளின் தொகுப்பு 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
இயற்கை எண்கள் N means { 1, 2, 3, …}, but see the article on natural numbers for a different convention.  = {|a| : a ∈ , a ≠ 0}
N
எண்s

Z
நிறை எண்கள் means {..., −3, −2, −1, 0, 1, 2, 3, …} and + means {1, 2, 3, …} = .  = {p, -p : p ∈ } ∪ {0}
Z
எண்s

Q
விகிதமுறு எண்கள் means {p/q : p ∈ , q ∈ }. 3.14000... ∈

π
Q
எண்s

R
மெய்யெண்s ℝ means the set of real numbers. π ∈

√(−1)
R
எண்s

C
சிக்கலெண்s means {a + b i : a,b ∈ }. i = √(−1) ∈
C
எண்s
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, where F'(x) = f(x)
C
தொகையீடு
𝕂

K
real or சிக்கலெண்s K means the statement holds substituting K for R and also for C.
$x^2\in\mathbb{C}\,\forall x\in \mathbb{K}$

because

$x^2\in\mathbb{C}\,\forall x\in \mathbb{R}$

and

$x^2\in\mathbb{C}\,\forall x\in \mathbb{C}$.
K
நேரியல் இயற்கணிதம்
எண்ணிலி ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. $\lim_{x\to 0} \frac{1}{|x|} = \infty$
எண்ணிலி
எண்s
||…||
norm || x || is the norm of the element x of a normed vector space. || x  + y || ≤  || x ||  +  || y ||
norm of

length of
நேரியல் இயற்கணிதம்
summation

$\sum_{k=1}^{n}{a_k}$ means a1 + a2 + … + an.

$\sum_{k=1}^{4}{k^2}$ = 12 + 22 + 32 + 42

= 1 + 4 + 9 + 16 = 30
sum over … from … to … of
எண்கணிதம்
product

$\prod_{k=1}^na_k$ means a1a2···an.

$\prod_{k=1}^4(k+2)$ = (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360
product over … from … to … of
எண்கணிதம்
கார்ட்டீசியன் பெருக்கற்பலன்

$\prod_{i=0}^{n}{Y_i}$ means the set of all (n+1)-tuples

(y0, …, yn).

$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$

the Cartesian product of; the direct product of
set theory
coproduct
coproduct over … from … to … of
category theory

வகையிடல் f ′(x) is the derivative of the function f at the point x, i.e., the சாய்வு of the தொடுகோடு to f at x.

The dot notation indicates a time derivative. That is $\dot{x}(t)=\frac{\partial}{\partial t}x(t)$.

If f(x) := x2, then f ′(x) = 2x
… prime

derivative of
நுண்கணிதம்
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
நுண்கணிதம்
தொகையீடு ab f(x) dx means the signed பரப்பளவு 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
நுண்கணிதம்
contour integral or closed line integral Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol .

The contour integral can also frequently be found with a subscript capital letter C, ∮C, denoting that a closed loop integral is, in fact, around a contour C, or sometimes dually appropriately, a circle C. In representations of Gauss's Law, a subscript capital S, ∮S, is used to denote that the integration is over a closed surface.

contour integral of
நுண்கணிதம்
gradient f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
vector calculus
விரிதல் (திசையன் நுண்கணிதம்) $\nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z}$ If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$, then $\nabla \cdot \vec v = 3y + 2yz$.
del dot, divergence of
vector calculus
curl $\nabla \times \vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i} + \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k}$ If $\vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k}$, then $\nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k}$.
curl of
vector calculus
partial differential 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, d
நுண்கணிதம்
boundary M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
இடவியல்
செங்குத்து 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
வடிவவியல்
bottom element x = ⊥ means x is the smallest element. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
||
சமாந்தரம் x || y means x is parallel to y. If l || m and m ⊥ n then l ⊥ n.
is parallel to
வடிவவியல்
entailment A  B means the sentence A entails the sentence B, that is in 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. A → B  ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
இயல்நிலை உட்குலம் N  G means that N is a normal subgroup of group G. Z(G G
is a normal subgroup of
குலக் கோட்பாடு
/
ஈவு குலம் G / H means the quotient of group G modulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = வார்ப்புரு:0, ''b'', {a, b+a}, வார்ப்புரு:2''a'', ''b''+2''a''
mod
குலக் கோட்பாடு
quotient set A/~ means the set of all ~ சமானப் பகுதிes in A. If we define ~ by x ~ y ⇔ x − y ∈ , then
/~ = {{x + n : n ∈ } : x ∈ (0,1])
mod
கணக் கோட்பாடு
approximately equal x ≈ y means x is approximately equal to y. π ≈ 3.14159
is approximately equal to
everywhere
isomorphism G ≈ H means that group G is isomorphic to group H. Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the கிளைன் நான்குறுப்புக்குலம்.
is isomorphic to
குலக் கோட்பாடு
~
same order of magnitude m ~ n means the quantities m and n have the same order of magnitude, or general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .)
2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10
roughly similar

poorly approximates
அண்ணளவாக்கக் கோட்பாடு
〈,〉

( | )

< , >

·

:
inner product x,y〉 means the inner product of x and y as defined in an inner product space.

For spatial vectors, the புள்ளிப் பெருக்கல் notation, x·y is common.
For matricies, the colon notation may be used.

The standard inner product between two vectors x = (2, 3) and y = (−1, 5) is:
〈x, y〉 = 2 × −1 + 3 × 5 = 13

$A:B = \sum_{i,j} A_{ij}B_{ij}$

inner product of
நேரியல் இயற்கணிதம்
tensor product V  U means the tensor product of V and U. {1, 2, 3, 4}  {1, 1, 2} =
வார்ப்புரு:1, 2, 3, 4, {1, 2, 3, 4}, வார்ப்புரு:2, 4, 6, 8
tensor product of
நேரியல் இயற்கணிதம்
*
convolution f * g means the convolution of f and g. $(f * g )(t) = \int f(\tau) g(t - \tau)\, d\tau$
convolution, convoluted with
functional analysis
$\bar{x}$
கூட்டுச்சராசரி $\bar{x}$ (often read as "x bar") is the கூட்டுச்சராசரி (average value of $x_i$). $x = \{1,2,3,4,5\}; \bar{x} = 3$.
overbar, … bar
புள்ளியியல்
$\overline{z}$
complex conjugate $\overline{z}$ is the complex conjugate of z. $\overline{3+4i} = 3-4i$
conjugate
சிக்கலெண்
$\triangleq$
delta equal to $\triangleq$ means equal by definition. When $\triangleq$ is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡. $p(x_1,x_2,...,x_n) \triangleq \prod_{i=1}^n p(x_i | x_{\pi_i})$.
equal by definition
everywhere