Use LEFT and RIGHT arrow keys to navigate between flashcards;
Use UP and DOWN arrow keys to flip the card;
H to show hint;
A reads text to speech;
93 Cards in this Set
 Front
 Back
 3rd side (hint)
= (1)

1)equality
is equal to; equals 
1)x = y means x and y represent the same thing or value.


≠

inequation
is not equal to; does not equal 
x ≠ y means that x and y do not represent the same thing or value.


<
> 
strict inequality
is less than, is greater than proper subgroup is a proper subgroup of 
x < y means x is less than y.
x > y means x is greater than y. H < G means H is a proper subgroup of G. 

≪
≫ 
(very) strict inequality
is much less than, is much greater than asymptotic comparison of smaller (greater) order than 
x ≪ y means x is much less than y.
x ≫ y means x is much greater than y. f ≪ g means the growth of f is asymptotically bounded by g. 2x ≪ x 

≤
≥ 
inequality
is less than or equal to, is greater than or equal to subgroup is a subgroup of 
x ≤ y means x is less than or equal to y.
x ≥ y means x is greater than or equal to y. H ≤ G means H is a subgroup of G. 

∝

proportionality
is proportional to; varies as 
y ∝ x means that y = kx for some constant k.
if y = 2x, then y ∝ x 

+

addition
plus; add disjoint union the disjoint union of ... and ... 
4 + 6 means the sum of 4 and 6.
A1 + A2 means the disjoint union of sets A1 and A2. 

− (3)

1)subtraction
minus; take; subtract 2)negative sign negative; minus; the opposite of 3)settheoretic complement minus; without 
9 − 4 means the subtraction of 4 from 9.
−3 means the negative of the number 3. A − B means the set that contains all the elements of A that are not in B 

×

multiplication
times; multiplied by Cartesian product the Cartesian product of ... and ...; the direct product of ... and ... cross product cross group of units the group of units of 
3 × 4 means the multiplication of 3 by 4.
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. u × v means the cross product of vectors u and v R× consists of the set of units of the ring R, along with the operation of multiplication. 

·

multiplication
times; multiplied by dot product dot 
3 · 4 means the multiplication of 3 by 4.
u · v means the dot product of vectors u and v 

÷
⁄ 
division
divided by; over quotient group mod quotient set mod 
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.
G / H means the quotient of group G modulo its subgroup H. A/~ means the set of all ~ equivalence classes in A. 

±

plusminus
plus or minus plusminus plus or minus 
6 ± 3 means both 6 + 3 and 6 − 3.
10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. 

∓

minusplus
minus or plus 
6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5).


√

square root
the (principal) square root of 
√x means the positive number whose square is x.


… (4)

1)absolute value or modulus
absolute value of; modulus of 2)Euclidean distance Euclidean distance between; Euclidean norm of 3)determinant determinant of 4)cardinality cardinality of; size of; order of 
1)x means the distance along the real line (or across the complex plane) between x and zero.
2)x – y means the Euclidean distance between x and y. 3)A means the determinant of the matrix A 4)X means the cardinality of the set X. 

… (2)

1)norm
norm of; length of 2)nearest integer function nearest integer to 
1) x  means the norm of the element x of a normed vector space.
2)x means the nearest integer to x, with halfintegers being rounded to even. 

∣ (3)
∤ 
1)divisor, divides
divides 2)conditional probability given 3)restriction restriction of … to …; restricted to 
1)ab means a divides b.
a∤b means a does not divide b. 2)P(AB) means the probability of the event a occurring given that b occurs. 3)fA means the function f restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f. 

 (3)

1)parallel
is parallel to 2)incomparability is incomparable to 3)exact divisibility exactly divides 
1)x  y means x is parallel to y.
2)x  y means x is incomparable to y. 3)p^a  n means pa exactly divides n (i.e. p^a divides n but p^a+1 does not). 

# (2)
♯ 
1)cardinality
cardinality of; size of; order of 2)connected sum connected sum of; knot sum of; knot composition of 
1)#X means the cardinality of the set X.
2)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 

ℵ (1)

1)aleph number
aleph 
1)ℵα represents an infinite cardinality (specifically, the αth one, where α is an ordinal).


: (3)

1)such that
such that; so that 2)field extension extends; over 3)inner product of matrices inner product of 
1) : means “such that”, and is used in proofs and the setbuilder notation (described below).
2)K : F means the field K extends the field F. 3)A : B means the inner product of the matrices A and B. 

! (2)

1)factorial
factorial 2)logical negation not 
1)n! means the product 1 × 2 × ... × n.
2)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. 

~ (5)

1)probability distribution
has distribution 2)row equivalence is row equivalent to 3)same order of magnitude roughly similar; poorly approximates 4)asymptotically equivalent is asymptotically equivalent to 5)equivalence relation are in the same equivalence class 
1)X ~ D, means the random variable X has the probability distribution D.
2)A~B means that B can be generated by using a series of elementary row operations on A 3)m ~ n means the quantities m and n have the same order of magnitude, or general size. 4)lim(x→∞) f(n)/g(n) = 1 5)a ~ b means b ∈ [a] (and equivalently [b] ∈ a). 

≈ (2)

1)approximately equal
is approximately equal to 2)isomorphism is isomorphic to 
1)x ≈ y means x is approximately equal to y.
2)G ≈ H means that group G is isomorphic (structurally identical) to group H. 

≀ (1)

1)wreath product
wreath product of … by … 
1)A ≀ H means the wreath product of the group A by the group H.


◅ (2)

1)normal subgroup
is a normal subgroup of 2)ideal is an ideal of 
1)N ◅ G means that N is a normal subgroup of group G.
2)I ◅ R means that I is an ideal of ring R. 

⋉ (1)
⋊ 
1)semidirect product
the semidirect product of 
1)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.


∴ (1)

1)therefore
therefore; so; hence 
1)Sometimes used in proofs before logical consequences.


∵ (1)

1)because
because; since 
1)Sometimes used in proofs before reasoning.


■ (1)
□ ∎ ▮ ‣ 
1)end of proof
QED; tombstone; Halmos symbol 
1)Used to mark the end of a proof.


⇒ (1)
→ ⊃ 
1)material implication
implies; if … then 
1)A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.


⇔ (1)
↔ 
1)material equivalence
if and only if; iff 
1)A ⇔ B means A is true if B is true and A is false if B is false.


¬ (1)
˜ 
1)logical negation
not 
1)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. 

∧ (2)

1)logical conjunction or meet in a lattice
and; min; meet 2)wedge product wedge product; exterior product 
1)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)). 2)u ∧ v means the wedge product of vectors u and v. This generalizes the cross product to higher dimensions. 

∨ (1)

1)logical disjunction or join in a lattice
or; max; join 
1)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)). 

⊕ (2)
⊻ 
1)exclusive or
xor 2)direct sum direct sum of 
1)The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.
2)The direct sum is a special way of combining several modules into one general module. 

∀ (1)

1)universal quantification
for all; for any; for each 
1)∀ x: P(x) means P(x) is true for all x.


∃ (1)

1)existential quantification
there exists; there is; there are 
1)∃ x: P(x) means there is at least one x such that P(x) is true.


∃! (1)

1)uniqueness quantification
there exists exactly one 
1)∃! x: P(x) means there is exactly one x such that P(x) is true.


:= (1)
≡ :⇔ ≜ ≝ ≐ 
1)definition
is defined as; equal by definition 
1)x := y or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.


≅ (2)

1)congruence
is congruent to 2)isomorphic is isomorphic to 
1)△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
2)G ≅ H means that group G is isomorphic (structurally identical) to group H. 

≡ (1)

1)congruence relation
... is congruent to ... modulo ... 
1)a ≡ b (mod n) means a − b is divisible by n


{ , } (1)

1)set brackets
the set of … 
1){a,b,c} means the set consisting of a, b, and c.


{ : } (1)
{  } 
1)set builder notation
the set of … such that 
1){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)}.


∅ (1)
{ } 
1)empty set
the empty set 
1)∅ means the set with no elements. { } means the same.


∈ (1)
∉ 
1)set membership
is an element of; is not an element of 
1)a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S.


⊆ (1)
⊂ 
1)subset
is a subset of 
1)(subset) A ⊆ B means every element of A is also an element of B.
(proper subset) A ⊂ B means A ⊆ B but A ≠ B. 

⊇ (1)
⊃ 
1)superset
is a superset of 
1)A ⊇ B means every element of B is also an element of A.
A ⊃ B means A ⊇ B but A ≠ B. 

∪ (1)

1)settheoretic union
the union of … or …; union 
1)A ∪ B means the set of those elements which are either in A, or in B, or in both.


∩ (1)

1)settheoretic intersection
intersected with; intersect 
1)A ∩ B means the set that contains all those elements that A and B have in common.


∆ (1)

1)symmetric difference
symmetric difference 
1)A ∆ B means the set of elements in exactly one of A or B.
(Not to be confused with delta, Δ, described below.) 

∖ (1)

1)settheoretic complement
minus; without 
1)A ∖ B means the set that contains all those elements of A that are not in B.


→ (1)

1)function arrow
from … to 
1)f: X → Y means the function f maps the set X into the set Y.


↦ (1)

1)function arrow
maps to 
1)f: a ↦ b means the function f maps the element a to the element b.


∘ (1)

1)function composition
composed with 
1)f∘g is the function, such that (f∘g)(x) = f(g(x)).


ℕ (1)
N 
1)natural numbers
N; the (set of) natural numbers 
1)N means either { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.


ℤ (1)
Z 
1)integers
Z; the (set of) integers 
1)ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...}.
ℤ+ or ℤ> means {1, 2, 3, ...} . ℤ≥ means {0, 1, 2, 3, ...} . 

ℤn (2)
ℤp Zn Zp 
1)integers mod n
Zn; the (set of) integers modulo n 2)padic integers the (set of) padic integers 
1)ℤn means {[0], [1], [2], ...[n−1]} with addition and multiplication modulo n.
2)Note that any letter may be used instead of p, such as n or l. 

ℚ (1)
Q 
1)rational numbers
Q; the (set of) rational numbers; the rationals 
1)ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}


ℝ (1)
R 
1)real numbers
R; the (set of) real numbers; the reals 
1)ℝ means the set of real numbers.


ℂ (1)
C 
1)complex numbers
C; the (set of) complex numbers 
1)ℂ means {a + b i : a,b ∈ ℝ}.



1)real or complex numbers
K 
1)K means both R and C: a statement containing K is true if either R or C is substituted for the K.


∞ (1)

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


⌊…⌋ (1)

1)floor
floor; greatest integer; entier 
1)⌊x⌋ means the floor of x, i.e. the largest integer less than or equal to x.


⌈…⌉ (1)

1)ceiling
ceiling 
1)⌈x⌉ means the ceiling of x, i.e. the smallest integer greater than or equal to x.


⌊…⌉ (1)

1)nearest integer function
nearest integer to 
1)⌊x⌉ means the nearest integer to x, with halfintegers being rounded to even.


[ : ] (1)

1)degree of a field extension
the degree of 
1)[K : F] means the degree of the extension K : F.


[ ] (8)
[ , ] [ , , ] 
1)equivalence class
the equivalence class of 2)floor floor; greatest integer; entier 3)nearest integer function nearest integer to 4)Iverson bracket 1 if true, 0 otherwise 5)image image of … under … 6)closed interval closed interval 7)commutator the commutator of 8)triple scalar product the triple scalar product of 
1)[a] means the equivalence class of a, i.e. {x : x ~ a}, where ~ is an equivalence relation.
2)[x] means the floor of x, i.e. the largest integer less than or equal to x. 3)[x] means the nearest integer to x, with halfintegers being rounded to even. 4)[S] maps a true statement S to 1 and a false statement S to 0. 5)f[X] means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ dom(f). 6) [a,b] = {x ∈ ℝ : a ≤ x ≤ b} 7)[g, h] = g−1h−1gh (or ghg−1h−1), if g, h ∈ G (a group). 8)[a, b, c] = a × b · c, the scalar product of a × b with c. 

( ) (5)
( , ) 
1)function application
of 2)image image of … under … 3)precedence grouping parentheses 4)tuple tuple; ntuple; ordered pair/triple/etc; row vector; sequence 5)highest common factor highest common factor; greatest common divisor; hcf; gcd 
1)f(x) means the value of the function f at the element x.
2)f(X) means { f(x) : x ∈ X }, the image of the function f under the set X ⊆ dom(f). 3)Perform the operations inside the parentheses first. 4)An ordered list (or sequence, or horizontal vector, or row vector) of values. 5)(a, b) means the highest common factor of a and b. 

( , ) (1)
] , [ 
1)open interval
open interval 
1)(a,b) = {x ∈ ℝ : a < x < b}


( , ] (1)
] , ] 
1)leftopen interval
halfopen interval; leftopen interval 
1)(a,b] = {x ∈ ℝ : a < x ≤ b}


[ , ) (1)
[ , [ 
1)rightopen interval
halfopen interval; rightopen interval 
1)[a,b) = {x ∈ ℝ : a ≤ x < b}


〈〉(4)
<> 〈,〉 <,> 
1)inner product
inner product of 2)linear span (linear) span of; linear hull of 3)subgroup generated by a set the subgroup generated by 4)tuple tuple; ntuple; ordered pair/triple/etc; row vector; sequence 
1)〈u,v〉 means the inner product of u and v, where u and v are members of an inner product space.
2)〈S〉 means the span of S ⊆ V. That is, it is the intersection of all subspaces of V which contain S. 〈u1, u2, …〉is shorthand for 〈{u1, u2, …}〉. 3)〈S〉 means the smallest subgroup of G (where S ⊆ G, a group) containing every element of S. 〈g1, g2, …〉is shorthand for 〈{g1, g2, …}〉 4)An ordered list (or sequence, or horizontal vector, or row vector) of values. 

〈〉 (1)
<> () 
1)inner product
inner product of 
1)〈u  v〉 means the inner product of u and v, where u and v are members of an inner product space. (u  v) means the same.


∑ (1)

1)summation
sum over … from … to … of 
1) ∑ak means a1 + a2 + … + an.


∏ (2)

1)product
product over … from … to … of 2)Cartesian product the Cartesian product of; the direct product of 
1) means a1a2···an.
2)means the set of all (n+1)tuples (y0, …, yn). 

∐ (1)

1)coproduct
coproduct over … from … to … of 
1)A general construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism.


′ (1)
• 
1)derivative
… prime derivative of 
1)f ′(x) means the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.


∫ (3)

1)indefinite integral or antiderivative
indefinite integral of the antiderivative of 2)definite integral integral from … to … of … with respect to 3)line integral line/path/curve integral of … along … 
1)∫ f(x) dx means a function whose derivative is f.
2)∫ab f(x) dx means the signed area between the xaxis and the graph of the function f between x = a and x = b. 3) 

∮ (1)

1)contour integral or closed line integral
contour integral of 
1)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 ∰.


∇ (3)

1)gradient
del, nabla, gradient of 2)divergence del dot, divergence of 3)curl curl of 
1)∇f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn).
2) 3) 

∂ (3)

1)partial derivative
partial, d 2)boundary boundary of 3)degree of a polynomial degree of 
1)∂f/∂xi means the partial derivative of f with respect to xi, where f is a function on (x1, …, xn).
2)∂M means the boundary of M 3)∂f means the degree of the polynomial f. 

Δ (1)

1)delta
delta; change in 
1)Δx means a (noninfinitesimal) change in x.


δ (2)

1)Dirac delta function
Dirac delta of 2)Kronecker delta Kronecker delta of 
1)δ(x)
2)δij 

<: (2)
<· 
1)cover
is covered by 2)subtype is a subtype of 
1)x <• y means that x is covered by y.
2)T1 <: T2 means that T1 is a subtype of T2. 

T (1)

1)T

1)AT means A, but with its rows swapped for columns.


⊤ (2)

1)top element
the top element 2)top type the top type; top 
1)⊤ means the largest element of a lattice.
2)⊤ means the top or universal type; every type in the type system of interest is a subtype of top. 

⊥ (6)

1)perpendicular
is perpendicular to 2)orthogonal complement orthogonal/perpendicular complement of; perp 3)coprime is coprime to 4)bottom element the bottom element 5)bottom type the bottom type; bot 6)comparability is comparable to 
1)x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.
2)W⊥ means the orthogonal complement of W (where W is a subspace of the inner product space V), the set of all vectors in V orthogonal to every vector in W. 3)x ⊥ y means x has no factor in common with y. 4)⊥ means the smallest element of a lattice. 5)⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. 6)x ⊥ y means that x is comparable to y. 

⊧ (1)

1)entailment
entails 
1)A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true.


⊢ (1)

1)inference
infers; is derived from 
1)x ⊢ y means y is derivable from x.


⊗ (1)

1)tensor product, tensor product of modules
tensor product of 
1) V ⊗ U means the tensor product of V and U. means the tensor product of modules V and U over the ring R.


* (3)

1)convolution
convolution, convolved with 2)complex conjugate conjugate 3)group of units the group of units of 
1)f * g means the convolution of f and g.
2)z* means the complex conjugate of z. 3)R* consists of the set of units of the ring R, along with the operation of multiplication. 

_
x (4) 
1)mean
overbar, … bar 2)complex conjugate conjugate 3)algebraic closure algebraic closure of 4)topological closure (topological) closure of 
1) (often read as “x bar”) is the mean (average value of xi).
2)means the complex conjugate of z. 3) is the algebraic closure of the field F. 4)is the topological closure of the set S. 