கணிதக் குறியீடுகள்

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	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
	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	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
	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 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 இயல் எண்.
	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 இயல் எண்.
	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
:= ≡ :⇔	வரைவிலக்கணம்	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
	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.	ℕ = { 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 ∈ 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 ∈ 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 ∈ 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
	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 ℝ ⊃ ℚ
	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 ∈ ℝ : x² = 1} ∩ ℕ = {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: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
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 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}
	N
	எண்s
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}
	Z
	எண்s
ℚ Q	விகிதமுறு எண்கள்	ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.	3.14000... ∈ ℚ π ∉ ℚ
	Q
	எண்s
ℝ R	மெய்யெண்s	ℝ means the set of real numbers.	π ∈ ℝ √(−1) ∉ ℝ
	R
	எண்s
ℂ C	சிக்கலெண்s	ℂ means {a + b i : a,b ∈ ℝ}.	i = √(−1) ∈ ℂ
	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 + … + a_n.	$\sum _{k=1}^{4}{k^{2}}$ = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
	sum over … from … to … of
	எண்கணிதம்
∏	product	$\prod _{k=1}^{n}a_{k}$ 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
	எண்கணிதம்
	கார்ட்டீசியன் பெருக்கற்பலன்	$\prod _{i=0}^{n}{Y_{i}}$ 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
	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	∫_a^b f(x) dx means the signed பரப்பளவு 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
	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, …, 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)
	டெல் இயக்கி, nabla, gradient of
	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, …, 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
	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 ∈ ℤ, then ℝ/~ = {{x + n : n ∈ ℤ} : 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
${\bar {x}}$ x̄	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