kanitha kuriyitukal (கண&#3007;தக&#3021; க&#3009;ற&#3007;ய&#3008;ட&#3009;கள&#3021;)

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 ...

−

கழித்தல்

9 − 4 means the subtraction of 4 from 9.

8 − 3 = 5

minus

negative sign

−3 means the negative of the number 3.

−(−5) = 5

negative; minus

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

பெருக்கல்

3 × 4 means the multiplication of 3 by 4.

7 × 8 = 56

times

Cartesian product

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 ...

cross product

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

dot product

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

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

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

measurement

∓

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

√

வர்க்கமூலம்

√x means 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(i φ/2).

√(-1) = i

the complex square root of …

square root

complex numbers

|…|

absolute value 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

propositional logic, Heyting algebra

Euclidean distance

|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

Geometry

Determinant

|A| means the determinant of the matrix A

$\begin{vmatrix} 1&2 \\ 2&4 \\ \end{vmatrix} = 0$

determinant of

Matrix theory

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

factorial

n ! is the product 1 × 2× ... × n.

4! = 1 × 2 × 3 × 4 = 24

factorial

combinatorics

transpose

Swap rows for columns

A i j = (A T) j i

transpose

matrix operations

probability distribution

X ~ D, means the random variable X has the probability distribution D.

X ~ N(0,1), the standard normal distribution

has distribution

statistics

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

Matrix theory

⇒

→

⊃

material implication

A ⇒ B 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

⇔

↔

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 A ∧ B 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 natural number.

and; min

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.

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 natural number.

or; max

propositional logic, lattice theory

⊕

⊻

exclusive 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

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 = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)

direct sum of

Abstract algebra

∀

universal quantification

∀ x: P(x) means P(x) is true for all x.

∀ n ∈ 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: 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: n + 5 = 2n.

there exists exactly one

predicate logic

:=

≡

:⇔

definition

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

(Some writers use ≡ to 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

geometry

≡

congruence relation

a ≡ b (mod n) means a − b is divisible by n

5 ≡ 11 (mod 3)

... is congruent to ... modulo ...

modular arithmetic

{ , }

தொடை brackets

{a,b,c} means the set consisting of a, b, and c.

ℕ = { 1, 2, 3, …}

the set of …

{ : }

{ | }

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 ∈ ℕ : n² < 20} = { 1, 2, 3, 4}

the set of … such that

∅

{ }

சூனியத்தொடை

∅ means the set with no elements. { } means the same.

{n ∈ ℕ : 1 < n² < 4} = ∅

the empty set

∈

∉

set membership

a ∈ S என்பது a , Sதொடையின் மூலகமாகும் ; a ∉ S என்பது a ,Sதொடையின் மூலகமல்ல என்றும் குறித்து நிற்கும் .

(1/2)⁻¹ ∈ ℕ

2⁻¹ ∉ ℕ

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

ℕ ⊂ ℚ

ℚ ⊂ ℝ

is a subset of

⊇

⊃

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

ℝ ⊃ ℚ

is a superset of

∪

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-theoretic intersection

∩

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

Δ

symmetric difference

A Δ B

means the set of elements in exactly one of A or B.

{1,5,6,8}

Δ

{2,5,8} = {1,2,6}

symmetric difference

∖

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

( )

function application

f(x) means the value of the function f at the element x.

If f(x) := x², then 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.

parentheses

everywhere

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

set theory,type theory

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

ℕ

N

இயற்கை எண்கள்

N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.

ℕ = {|a| : a ∈ ℤ, a ≠ 0}

ℤ

Z

நிறை எண்கள்

ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ⁺ means {1, 2, 3, ...} = ℕ.

ℤ = {p, -p : p ∈ ℕ} ∪ {0}

ℚ

Q

விகிதமுறு எண்கள்

ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.

3.14000... ∈ ℚ

π ∉ ℚ

ℝ

R

real numbers

ℝ means the set of real numbers.

π ∈ ℝ

√(−1) ∉ ℝ

ℂ

C

complex numbers

ℂ means {a + b i : a,b ∈ ℝ}.

i = √(−1) ∈ ℂ

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)

integral calculus

𝕂

K

real or complex numbers

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

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

∞

infinity

∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.

$\lim_{x\to 0} \frac{1}{|x|} = \infty$

infinity

||…||

norm

|| x || is the norm of the element x of a normed vector space.

|| x + y || ≤ || x || + || y ||

norm of

length of

∑

summation

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

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

= 1 + 4 + 9 + 16 = 30

sum over … from … to … of

∏

product

$\prod_{k=1}^na_k$ means a₁a₂···a_n.

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

= 3 × 4 × 5 × 6 = 360

product over … from … to … of

Cartesian product

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

(y₀, …, y_n).

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

the Cartesian product of; the direct product of

∐

coproduct

coproduct over … from … to … of

category 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.
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

∫

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

definite integral

∫_a^b f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.

∫₀^b x² dx = b³/3;

integral from … to … of … with respect to

∮

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

∇

gradient

∇f (x₁, …, x_n) is the vector of partial derivatives (∂f / ∂x₁, …, ∂f / ∂x_n).

If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)

del, nabla, gradient of

vector calculus

divergence

$\nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z}$

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, …, x_n), ∂f/∂x_i is the derivative of f with respect to x_i, with all other variables kept constant.

If f(x,y) := x²y, then ∂f/∂x = 2xy

partial, d

propositional logic, predicate logic

boundary

∂M means the boundary of M

∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}

boundary of

topology

⊥

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

geometry

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

geometry

⊧

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

◅

normal subgroup

N ◅ G means that N is a normal subgroup of group G.

Z(G) ◅ G

is a normal subgroup of

group theory

quotient group

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}}

mod

group theory

quotient set

A/~ means the set of all ~ equivalence classes in A.

If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then
ℝ/~ = {{x + n : n ∈ ℤ} : x ∈ (0,1]}

mod

≈

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 Klein four-group.

is isomorphic to

group theory

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

Approximation theory

〈,〉

( | )

< , >

·

:

inner product

〈x,y〉 means the inner product of x and y as defined in an inner product space.
For spatial vectors, the dot product 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 =	∑	A_ijB_ij
	i,j

inner product of

⊗

tensor product

V ⊗ U means 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