smcproject/ml-wiki-sentences huggingface dataset:
if X is a uniformly random day of the year P (X is May 25 | X is in May) = 1/31 restriction restriction of … to …; restricted to set theory f | A means the function f restricted to the set A , that is, it is the function with domain A ∩ dom( f ) that agrees with f .
The function f : R → R defined by f ( x ) = x 2 is not injective, but f | R + is injective.
such that such that; so that എല്ലായിടത്തും | means “such that”, see ":" ( described below ).
S = {(x,y) | 0 y f(x)} The set of (x,y) such that y is greater than 0 and less than f(x).
|| ‖ {\displaystyle \|\!\,} parallel is parallel to ജ്യാമിതി x || y means x is parallel to y .
If l || m and m ⊥ n then l ⊥ n .
incomparability is incomparable to order theory x || y means x is incomparable to y .
{1,2} || {2,3} under set containment.
exact divisibility exactly divides സംഖ്യാസിദ്ധാന്തം p a || n means p a exactly divides n (i.e. p a divides n but p a +1 does not).
2 3 || 360.
# # {\displaystyle \#\!\,} cardinality cardinality of; size of; order of set theory # X means the cardinality of the set X .
(|…| may be used instead as described above. ) #{4, 6, 8} = 3 connected sum connected sum of; knot sum of; knot composition of topology , knot theory A # B is the connected sum of the manifolds A and B .
If A and B are knots, then this denotes the knot sum, which has a slightly stronger condition.
A # S m is homeomorphic to A , for any manifold A , and the sphere S m .
ℵ ℵ {\displaystyle \aleph \!\,} aleph number aleph set theory ℵ α represents an infinite cardinality (specifically, the α -th one, where α is an ordinal).
|ℕ| = ℵ 0 , which is called aleph-null.
ℶ ℶ {\displaystyle \beth \!\,} beth number beth set theory ℶ α represents an infinite cardinality (similar to ℵ , but ℶ does not necessarily index all of the numbers indexed by ℵ . ).
ℶ 1 = | P ( N ) | = 2 ℵ 0 .
{\displaystyle \beth _{1}=|P(\mathbb {N} )|=2^{\aleph _{0}}.
} 𝔠 c {\displaystyle {\mathfrak {c}}\!
\,} cardinality of the continuum cardinality of the continuum; cardinality of the real numbers; c; set theory The cardinality of R {\displaystyle \mathbb {R} } is denoted by | R | {\displaystyle |\mathbb {R} |} or by the symbol c {\displaystyle {\mathfrak {c}}} (a lowercase Fraktur letter C).
c = ℶ 1 {\displaystyle {\mathfrak {c}}={\beth }_{1}} : : {\displaystyle :\!\,} such that such that; so that എല്ലായിടത്തും : means “such that”, and is used in proofs and the set-builder notation ( described below ).
∃ n ∈ ℕ: n is even.
field extension extends; over field theory K : F means the field K extends the field F .
This may also be written as K ≥ F .
ℝ : ℚ inner product of matrices inner product of linear algebra A : B means the Frobenius inner product of the matrices A and B .
The general inner product is denoted by ⟨ u , v ⟩, ⟨ u | v ⟩ or ( u | v ), as described below.
For spatial vectors, the dot product notation, x · y is common.
See also Bra-ket notation .
A : B = ∑ i , j A i j B i j {\displaystyle A:B=\sum _{i,j}A_{ij}B_{ij}\!
\,} index of a subgroup index of subgroup ഗ്രൂപ്പ് സിദ്ധാന്തം The index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" ( cosets ) of H that fill up G | G : H | = | G | | H | {\displaystyle |G:H|={\frac {|G|}{|H|}}} !
{\displaystyle !\!\,} ഫാക്റ്റോറിയൽ ഫാക്റ്റോറിയൽ combinatorics n !
means the product 1 × 2 × ...
× n .
= 1 × 2 × 3 × 4 = 24 logical n
egation not propositional logic 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 ! is primarily from computer science. It is avoided in mathematical texts, where the notation ¬ A is preferred. ) !
(! A ) ⇔ A x ≠ y ⇔ !
( x = y ) ~ ∼ {\displaystyle \sim \!\,} probability distribution has distribution സ്ഥിതിഗണിതം X ~ D , means the random variable X has the probability distribution D .
X ~ N (0,1), the standard normal distribution row equivalence is row equivalent to matrix theory A ~ B means that B can be generated by using a series of elementary row operations on A [ 1 2 2 4 ] ∼ [ 1 2 0 0 ] {\displaystyle {\begin{bmatrix}1 2\\2 4\\\end{bmatrix}}\sim {\begin{bmatrix}1 2\\0 0\\\end{bmatrix}}} same order of magnitude roughly similar; poorly approximates approximation theory 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 π 2 ≈ 10 asymptotically equivalent is asymptotically equivalent to asymptotic analysis f ~ g means lim n → ∞ f ( n ) g ( n ) = 1 {\displaystyle \lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1} .
x ~ x+1 equivalence relation are in the same equivalence class എല്ലായിടത്തും a ~ b means b ∈ [ a ] {\displaystyle b\in [a]} (and equivalently a ∈ [ b ] {\displaystyle a\in [b]} ).
1 ~ 5 mod 4 ≈ ≈ {\displaystyle \approx \!\,} approximately equal is approximately equal to എല്ലായിടത്തും x ≈ y means x is approximately equal to y .
π ≈ 3.14159 isomorphism is isomorphic to ഗ്രൂപ്പ് സിദ്ധാന്തം G ≈ H means that group G is isomorphic (structurally identical) to group H .
( ≅ can also be used for isomorphic, as described below. ) Q / {1, − 1} ≈ V , where Q is the quaternion group and V is the Klein four-group .
≀ ≀ {\displaystyle \wr \!\,} wreath product wreath product of … by … ഗ്രൂപ്പ് സിദ്ധാന്തം A ≀ H means the wreath product of the group A by the group H .
This may also be written A wr H .
S n ≀ Z 2 {\displaystyle S_{n}\wr Z_{2}} is isomorphic to the automorphism group of the complete bipartite graph on ( n , n ) vertices.
◅ ▻ ◃ {\displaystyle \triangleleft \!\,} ▹ {\displaystyle \triangleright \!\,} normal subgroup is a normal subgroup of ഗ്രൂപ്പ് സിദ്ധാന്തം N ◅ G means that N is a normal subgroup of group G .
Z ( G ) ◅ G ideal is an ideal of ring theory I ◅ R means that I is an ideal of ring R .
(2) ◅ Z antijoin the antijoin of relational algebra R ▻ S means the antijoin of the relations R and S , the tuples in R for which there is not a tuple in S that is equal on their common attribute names.
R ▹ {\displaystyle \triangleright } S = R - R ⋉ {\displaystyle \ltimes } S ⋉ ⋊ ⋉ {\displaystyle \ltimes \!\,} ⋊ {\displaystyle \rtimes \!\,} semidirect product the semidirect product of ഗ്രൂപ്പ് സിദ്ധാന്തം N ⋊ φ H is the semidirect product of N (a normal subgroup) and H (a subgroup), with respect to φ.
Also, if G = N ⋊ φ H , then G is said to split over N .
( ⋊ may also be written the other way round, as ⋉ , or as ×.) D 2 n ≅ C n ⋊ C 2 {\displaystyle D_{2n}\cong C_{n}\rtimes C_{2}} semijoin the semijoin of relational algebra R ⋉ S is the semijoin of the relations R and S , the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names.
R ⋉ {\displaystyle \ltimes } S = Π {\displaystyle \Pi } a 1 ,..
, a n ( R ⋈ {\displaystyle \bowtie } S ) ⋈ ⋈ {\displaystyle \bowtie \!\,} natural join the natural join of relational algebra R ⋈ S is the natural join of the relations R and S , the set of all combinations of tuples in R and S that are equal on their common attribute names.
∴ ∴ {\displaystyle \therefore \!\,} therefore therefore; so; hence എല്ലായിടത്തും Sometimes used in proofs before logical consequences .
Socrates is a human.
∴ Socrates is mortal.
∵ ∵ {\displaystyle \because \!\,} because because; since എല്ലായിടത്തും Sometimes used in proofs before reasoning.
3331 is prime ∵ it has no positive integer factors other than itself and one.
■ □ ∎ ▮ ‣ ◼ {\displaystyle \blacksquare \!\,} ◻ {\displaystyle \Box \!\,} ▸ {\displaystyle \blacktriangleright \!\,} end of proof QED ; tombstone ; Halmos symbol എല്ലായിടത്തും Used to mark the end of a proof.
( May also be written Q.E.D.) ⇒ → ⊃ ⇒ {\displaystyle \Rightarrow \!\,} → {\displaystyle \rightarrow \!\,} ⊃ {\displaystyle \supset \!\,} material implication implies; if … then propositional logic , Heyting algebra 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 ഫലനംs given below. ) (⊃ may mean the same as ⇒ , or it may have the meaning for superset given below. ) x = 2 ⇒ x 2 = 4 is true, but x 2 = 4 ⇒ x = 2 is in general false (since x could be − 2).
⇔ ↔ ⇔ {\displaystyle \Leftrightarrow \!\,} ↔ {\displaystyle \leftrightarrow \!\,} material equivalence if and only if; iff propositional logic 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 ¬ ˜ ¬ {\displaystyle \neg \!\,} ∼ {\displaystyle \sim \!\,} logical negation not propositional logic 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. Computer scientists will often use ! but this is avoided in mathematical texts. ) ¬(¬ A ) ⇔ A x ≠ y ⇔ ¬( x = y ) ∧ ∧ {\displaystyle \land \!\,} logical conjunction or meet in a lattice and; min; meet propositional logic , lattice theory 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 ⇔ n = 3 when n is a എണ്ണൽ സംഖ്യ .
wedge product wedge product; exterior product linear algebra u ∧ v means the wedge product of vectors u and v .
This generalizes the cross product to higher dimensions.
( For vectors in R 3 , × can also be used. ) u ∧ v = u × v , if u , v ∈ R 3 {\displaystyle u\wedge v=u\times v,{\mbox{ if }}u,v\in \mathbb {R} ^{3}} exponentiation … (raised) to the power of … എല്ലായിടത്തും a ^ b means a raised to the power of b ( a ^ b is more commonly written a b . The symbol ^ is generally used in programming languages where ease of typing and use of plain ASCII text is preferred. ) 2^3 = 2 3 = 8 ∨ ∨ {\displaystyle \lor \!\,} logical disjunction or join in a lattice or; max; join propositional logic , lattice theory 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 എണ്ണൽ സംഖ്യ .
⊕ ⊻ ⊕ {\displaystyle \oplus \!\,} ⊻ {\displaystyle \veebar \!\,} exclusive or xor propositional logic , Boolean algebra 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.
direct sum direct sum of അമൂർത്തബീജഗണിതം The direct sum is a special way of combining several objects into one general object.
( The bun symbol ⊕, or the coproduct 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 = {0}) ∀ ∀ {\displaystyle \forall \!\,} universal quantification for all; for any; for each predicate logic ∀ x : P ( x ) means P ( x ) is true for all x .
∀ n ∈ ℕ : n 2 ≥ n .
∃ ∃ {\displaystyle \exists \!\,} existential quantification there exists; there is; there are predicate logic ∃ x : P ( x ) means there is at least one x such that P ( x ) is true.
∃ n ∈ ℕ : n is even.
∃ !
{\displaystyle \exists !\!\,} uniqueness quantification there exists exactly one predicate logic ∃ !
x : P ( x ) means there is exactly one x such that P ( x ) is true.
n ∈ ℕ : n + 5 = 2 n .
=: := ≡ :⇔ ≜ ≝ ≐ =: {\displaystyle =:\!\,} := {\displaystyle :=\!\,} ≡ {\displaystyle \equiv \!\,} :⇔ {\displaystyle :\Leftrightarrow \!\,} ≜ {\displaystyle \triangleq \!\,} = d e f {\displaystyle {\overset {\underset {\mathrm {def} }{}}{=}}\!
\,} ≐ {\displaystyle \doteq \!\,} definition is defined as; is equal by definition to എല്ലായിടത്തും x := y , y =: x or x ≡ y means x is defined to be another name for y , under certain assumptions taken in context.
( Some writers use ≡ to mean congruence ).
P :⇔ Q means P is defined to be logically equivalent to Q .
cosh ⁡ x := e x + e − x 2 {\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}} ≅ ≅ {\displaystyle \cong \!\,} congruence is congruent to ജ്യാമിതി △ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
isomorphic is isomorphic to അമൂർത്തബീജഗണിതം G ≅ H means that group G is isomorphic (structurally identical) to group H .
(≈ can also be used for isomorphic, as described above. ) R 2 ≅ C {\displaystyle \mathbb {R} ^{2}\cong \mathbb {C} } .
≡ ≡ {\displaystyle \equiv \!\,} congruence relation ...
is congruent to ...
modulo ...
modular arithmetic a ≡ b (mod n ) means a − b is divisible by n 5 ≡ 2 (mod 3) { , } { , } {\displaystyle {\{\ ,\!\ \}}\!
\,} ഗണം brackets the set of … set theory { a , b , c } means the set consisting of a , b , and c .
ℕ = { 1, 2, 3, …} { : } { | } { : } {\displaystyle \{\ :\ \}\!
\,} { | } {\displaystyle \{\ |\ \}\!
\,} set builder notation
the set of … such that set theory { 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 : n 2 20} = { 1, 2, 3, 4} ∅ { } ∅ {\displaystyle \emptyset \!\,} ∅ {\displaystyle \varnothing \!\,} { } {\displaystyle \{\}\!
\,} empty set the empty set set theory ∅ means the set with no elements.
{ } means the same.
{ n ∈ ℕ : 1 n 2 4} = ∅ ∈ ∉ ∈ {\displaystyle \in \!\,} ∉ {\displaystyle \notin \!\,} set membership is an element of; is not an element of എല്ലായിടത്തും, set theory a ∈ S means a is an element of the set S ; a ∉ S means a is not an element of S .
(1/2) − 1 ∈ ℕ 2 − 1 ∉ ℕ ⊆ ⊂ ⊆ {\displaystyle \subseteq \!\,} ⊂ {\displaystyle \subset \!\,} subset is a subset of set theory (subset) A ⊆ B means every element of A is also an 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 ⊇ ⊃ ⊇ {\displaystyle \supseteq \!\,} ⊃ {\displaystyle \supset \!\,} superset is a superset of set theory A ⊇ B means every element of B is also an 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 ∪ ∪ {\displaystyle \cup \!\,} set-theoretic union the union of … or …; union set theory A ∪ B means the set of those elements which are either in A , or in B , or in both.
A ⊆ B ⇔ ( A ∪ B ) = B ∩ ∩ {\displaystyle \cap \!\,} set-theoretic intersection intersected with; intersect set theory A ∩ B means the set that contains all  those elements that A and B have in common.
{ x ∈ ℝ : x 2 = 1} ∩ ℕ = {1} ∆ △ {\displaystyle \vartriangle \!\,} symmetric difference symmetric difference set theory A ∆ B means the set of elements in exactly one of A or B .
( Not to be confused with delta , Δ, described below. ) {1,5,6,8} ∆ {2,5,8} = {1,2,6} ∖ ∖ {\displaystyle \setminus \!\,} set-theoretic complement minus; without set theory 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} → → {\displaystyle \to \!\,} ഫലനം arrow from … to set theory , type theory f : X → Y means the function f maps the set X into the set Y .
Let f : ℤ → ℕ ∪{0} be defined by f ( x ) := x 2 .
↦ ↦ {\displaystyle \mapsto \!\,} ഫലനം arrow maps to set theory f : a ↦ b means the function f maps the element a to the element b .
Let f : x ↦ x +1 (the successor function).
∘ ∘ {\displaystyle \circ \!\,} function composition composed with set theory f ∘ g is the function, such that ( f ∘ g )( x ) = f ( g ( x )).
if f ( x ) := 2 x , and g ( x ) := x + 3, then  ( f ∘ g )( x ) = 2( x + 3).
ℕ N N {\displaystyle \mathbb {N} \!
\,} N {\displaystyle \mathbf {N} \!
\,} എണ്ണൽ സംഖ്യകൾ N; the (set of) natural numbers സംഖ്യകൾ N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.
The choice depends on the area of mathematics being studied; e.g. number theorists prefer the latter; analysts , set theorists and computer scientists prefer the former.
To avoid confusion, always check an author's definition of N .
Set theorists often use the notation ω (for least infinite ordinal ) to denote the set of natural numbers (including zero), along with the standard ordering relation ≤.
ℕ = {| a | : a ∈ ℤ } or ℕ = {| a | 0: a ∈ ℤ } ℤ Z Z {\displaystyle \mathbb {Z} \!
\,} Z {\displaystyle \mathbf {Z} \!
\,} പൂർണ്ണസംഖ്യകൾ Z; the (set of) integers സംഖ്യകൾ ℤ means {..., − 3, − 2, − 1, 0, 1, 2, 3, ...}.
ℤ + or ℤ means {1, 2, 3, ...} .
ℤ * or ℤ ≥ means {0, 1, 2, 3, ...} .
ℤ = { p , − p : p ∈ ℕ ∪ {0} ​ } ℤ n ℤ p Z n Z p Z n {\displaystyle \mathbb {Z} _{n}\!
\,} Z p {\displaystyle \mathbb {Z} _{p}\!
\,} Z n {\displaystyle \mathbf {Z} _{n}\!
\,} Z p {\displaystyle \mathbf {Z} _{p}\!
\,} integers mod n Z n ; the (set of) integers modulo n സംഖ്യകൾ ℤ n means {[0], [1], [2], ...
[ n − 1]} with addition and multiplication modulo n .
Note that any letter may be used instead of n , such as p .
To avoid confusion with p-adic numbers, use ℤ / p ℤ or ℤ /( p ) instead.
ℤ 3 = {[0], [1], [2]} p -adic integers the (set of) p -adic integers സംഖ്യകൾ Note that any letter may be used instead of p , such as n or l .
ℙ P P {\displaystyle \mathbb {P} \!
\,} P {\displaystyle \mathbf {P} \!
\,} projective space P; the projective space, the projective line, the projective plane topology ℙ means a space with a point at infinity.
P 1 {\displaystyle \mathbb {P} ^{1}} , P 2 {\displaystyle \mathbb {P} ^{2}} probability the probability of probability theory ℙ ( X ) means the probability of the event X occurring.
This may also be written as P( X ),  Pr( X ), P[ X ] or Pr[ X ].
If a fair coin is flipped, ℙ (Heads) = ℙ (Tails) = 0.5.
ℚ Q Q {\displaystyle \mathbb {Q} \!
\,} Q {\displaystyle \mathbf {Q} \!
\,} ഭിന്നകങ്ങൾ Q; the (set of) rational numbers; ഭിന്നകങ്ങൾ സംഖ്യകൾ ℚ എന്നാൽ { p / q : p ∈ ℤ , q ∈ ℕ }.
3.14000...
∈ ℚ π ∉ ℚ ℝ R R {\displaystyle \mathbb {R} \!
\,} R {\displaystyle \mathbf {R} \!
\,} വാസ്തവികസംഖ്യ R; the (set of) real numbers; the reals സംഖ്യകൾ ℝ means the set of real numbers.
π ∈ ℝ √( − 1) ∉ ℝ ℂ C C {\displaystyle \mathbb {C} \!
\,} C {\displaystyle \mathbf {C} \!
\,} മിശ്രസംഖ്യകൾ C; the (set of) complex numbers സംഖ്യകൾ ℂ means { a + b i : a , b ∈ ℝ }.
i = √( − 1) ∈ ℂ ℍ H H {\displaystyle \mathbb {H} \!
\,} H {\displaystyle \mathbf {H} \!
\,} quaternions or Hamiltonian quaternions H; the (set of) quaternions സംഖ്യകൾ ℍ means { a + b i + c j + d k : a , b , c , d ∈ ℝ }.
O O {\displaystyle O} Big O notation big-oh of Computational complexity theory The Big O notation describes the limiting behavior of a ഫലനം , when the argument tends towards a particular value or അനന്തത .
If f(x) = 6x 4 − 2x 3 + 5 and g(x) = x 4 , then f ( x ) = O ( g ( x ) ) as x → ∞ {\displaystyle f(x)=O(g(x)){\mbox{ as }}x\to \infty \,} ∞ ∞ {\displaystyle \infty \!\,} അനന്തത അനന്തത സംഖ്യകൾ ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits .
lim x → 0 1 | x | = ∞ {\displaystyle \lim _{x\to 0}{\frac {1}{|x|}}=\infty } ⌊…⌋ ⌊ … ⌋ {\displaystyle \lfloor \ldots \rfloor \!\,} floor floor; greatest integer; entier സംഖ്യകൾ ⌊ x ⌋ means the floor of x , i.e. the largest integer less than or equal to x .
( This may also be written [ x ], floor( x ) or int( x ).
) ⌊4⌋ = 4, ⌊2.1⌋ = 2, ⌊2.9⌋ = 2, ⌊ − 2.6⌋ = − 3 ⌈…⌉ ⌈ … ⌉ {\displaystyle \lceil \ldots \rceil \!\,} ceiling ceiling സംഖ്യകൾ ⌈ x ⌉ means the ceiling of x , i.e. the smallest integer greater than or equal to x .
( This may also be written ceil( x ) or ceiling( x ).
) ⌈4⌉ = 4, ⌈2.1⌉ = 3, ⌈2.9⌉ = 3, ⌈ − 2.6⌉ = − 2 ⌊…⌉ ⌊ … ⌉ {\displaystyle \lfloor \ldots \rceil \!\,} nearest integer function nearest integer to സംഖ്യകൾ ⌊ x ⌉ means the nearest integer to x .
( This may also be written [ x ], || x ||, nint( x ) or Round( x ).
) ⌊2⌉ = 2, ⌊2.6⌉ = 3, ⌊-3.4⌉ = -3, ⌊4.49⌉ = 4 [ : ] [ : ] {\displaystyle [\ :\ ]\!
\,} degree of a field extension the degree of field theory [ K : F ] means the degree of the extension K : F .
[ℚ(√2) : ℚ] = 2 [ℂ : ℝ] = 2 [ℝ : ℚ] = ∞ [ ] [ , ] [ , , ] [ ] {\displaystyle [\ ]\!
\,} [ , ] {\displaystyle [\ ,\ ]\!
\,} [ , , ] {\displaystyle [\ ,\ ,\ ]\!
\,} equivalence class the equivalence class of അമൂർത്തബീജഗണിതം [ a ] means the equivalence class of a , i.e. { x : x ~ a }, where ~ is an equivalence relation .
[ a ] R means the same, but with R as the equivalence relation.
Let a ~ b be true iff a ≡ b ( mod 5).
Then [2] = {…, − 8, − 3, 2, 7, …}.
floor floor; greatest integer; entier സംഖ്യകൾ [ x ] means the floor of x , i.e. the largest integer less than or equal to x .
( This may also be written ⌊ x ⌋, floor( x ) or int( x ).
Not to be confused with the nearest integer function, as described below.
) [3] = 3, [3.5] = 3, [3.99] = 3, [ − 3.7] = − 4 nearest integer function nearest integer to സംഖ്യകൾ [ x ] means the nearest integer to x .
( This may also be written ⌊ x ⌉, || x ||, nint( x ) or Round( x ).
Not to be confused with the floor function, as described above.
) [2] = 2, [2.6] = 3, [-3.4] = -3, [4.49] = 4 Iverson bracket 1 if true, 0 otherwise propositional logic [ S ] maps a true statement S to 1 and a false statement S to 0.
[0=5]=0, [7 ∈ {2,3,4}]=1, [5 ∈ {2,3,4}]=0 image image of … under … എല്ലായിടത്തും f [ X ] means { f ( x ) : x ∈ X }, the image of the function f under the set X ⊆ dom ( f ).
( This may also be written as f ( X ) if there is no risk of confusing the image of f under X with the function application f of X .
Another notation is Im f , the image of f under its domain.
) sin ⁡ [ R ] = [ − 1 , 1 ] {\displaystyle \sin[\mathbb {R} ]=[-1,1]} closed interval closed interval order theory [ a , b ] = { x ∈ R : a ≤ x ≤ b } {\displaystyle [a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}} .
0 and 1/2 are in the interval [0,1].
commutator the commutator of ഗ്രൂപ്പ് സിദ്ധാന്തം , ring theory [ g , h ] = g − 1 h − 1 gh (or ghg − 1 h − 1 ), if g , h ∈ G (a group ).
[ a , b ] = ab − ba , if a , b ∈ R (a ring or commutative algebra ).
x y = x [ x , y ] (ഗ്രൂപ്പ് സിദ്ധാന്തം).
[ AB , C ] = A [ B , C ] + [ A , C ] B (ring theory).
triple scalar product the triple scalar product of vector calculus [ a , b , c ] = a × b · c , the scalar product of a × b with c .
[ a , b , c ] = [ b , c , a ] = [ c , a , b ].
( ) ( , ) ( ) {\displaystyle (\ )\!
\,} ( , ) {\displaystyle (\ ,\ )\!
\,} ഫലനം application of set theory f ( x ) means the value of the function f at the element x .
If f ( x ) := x 2 , then f (3) = 3 2 = 9.
image image of … under … എല്ലായിടത്തും f ( X ) means { f ( x ) : x ∈ X }, the image of the function f under the set X ⊆ dom ( f ).
( This may also be written as f [ X ] if there is a risk of confusing the image of f under X with the function application f of X .
) sin ⁡ ( R ) = [ − 1 , 1 ] {\displaystyle \sin(\mathbb {R} )=[-1,1]} combinations (from) n choose r combinatorics ( n r ) {\displaystyle {\begin{pmatrix}n\\r\end{pmatrix}}} means the number of combinations of r elements drawn from a set of n elements.
( This may also be written as n C r .) ( 5 2 ) = 10 {\displaystyle {\begin{pmatrix}5\\2\end{pmatrix}}=10} precedence grouping parentheses എല്ലായിടത്തും Perform the operations inside the parentheses first.
(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
tuple tuple; n -tuple; ordered pair/triple/etc; row vector; sequence എല്ലായിടത്തും An ordered list (or sequence, or horizontal vector, or row vector) of values.
( Note that the notation ( a , b ) is ambiguous: it could be an ordered pair or an open interval.
Set theorists and computer scientists often use angle brackets ⟨ ⟩ instead of parentheses.
) ( a , b ) is an ordered pair (or 2-tuple).
( a , b , c ) is an ordered triple (or 3-tuple).
( ) is the empty tuple (or 0-tuple).
highest common factor highest common factor; greatest common divisor; hcf; gcd സംഖ്യാസിദ്ധാന്തം ( a , b ) means the highest common factor of a and b .
( This may also be written hcf( a , b ) or gcd( a , b ).
) (3, 7) = 1 (ഇവ സഹ-അഭാജ്യങ്ങളാണ്); (15, 25) = 5.
( , ) ] , [ ( , ) {\displaystyle (\ ,\ )\!
\,} ] , [ {\displaystyle ]\ ,\ [\!\,} open interval open interval order theory ( a , b ) = { x ∈ R : a x b } {\displaystyle (a,b)=\{x\in \mathbb {R} :a x b\}} .
The notation ] a , b [ can be used instead. ) 4 is not in the interval (4, 18).
(0, +∞) equals the set of positive real numbers.
( , ] ] , ] ( , ] {\displaystyle (\ ,\ ]\!
\,} ] , ] {\displaystyle ]\ ,\ ]\!
\,} left-open interval half-open interval; left-open interval order theory ( a , b ] = { x ∈ R : a x ≤ b } {\displaystyle (a,b]=\{x\in \mathbb {R} :a x\leq b\}} .
( − 1, 7] and ( − ∞, − 1] [ , ) [ , [ [ , ) {\displaystyle [\ ,\ )\!