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Symbol  Name  reads as  Category 

+  addition  plus  arithmetic 
4 + 6 = 10 means that if four is added to 6, the sum, or result, is 10.  
43 + 65 = 108; 2 + 7 = 9
 
  subtraction  minus  arithmetic 
9  4 = 5 means that if 4 is subtracted from 9, the result will be 5. The  sign is unique in that it can also denote that a number is negative. For example, 5 + (3) = 2 means that if five and negative three are added, the result is two.  
87  36 = 51
 
⇒
 material implication  implies; if .. then  propositional logic 
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 mentioned further down  
x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2)
 
⇔
 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
 
∧  logical conjunction  and  propositional logic 
the statement A ∧ B is true if A and B are both true; else it is false  
n < 4 ∧ n > 2 ⇔ n = 3 when n is a natural number
 
∨  logical disjunction  or  propositional logic 
the statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false  
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number
 
¬
 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  
¬(A ∧ B) ⇔ (¬A) ∨ (¬B); x ∉ S ⇔ ¬(x ∈ S)
 
∀  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  
∃  existential quantification  there exists  predicate logic 
∃ x: P(x) means: there is at least one x such that P(x) is true  
∃ n ∈ N: n + 5 = 2n
 
=  equality  equals  everywhere 
x = y means: x and y are different names for precisely the same thing  
1 + 2 = 6 − 3
 
:=
 definition  is defined as  everywhere 
x := y means: x is defined to be another name for y 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)
 
{ , }  set brackets  the set of ...  set theory 
{a,b,c} means: the set consisting of a, b, and c  
N = {0,1,2,...}
 
{ : }
 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} = {0,1,2,3,4}
 
∅
 empty set  empty set  set theory 
{} means: the set with no elements; ∅ is the same thing  
{n ∈ N : 1 < n^{2} < 4} = {}  
∈
 set membership  in; is in; is an element of; is a member of; belongs to  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
 
⊆
 subset  is a subset of  set theory 
A ⊆ B means: every element of A is also element of B A ⊂ B means: A ⊆ B but A ≠ B  
A ∩ B ⊆ A; Q ⊂ R
 
∪  set theoretic union  the union of ... and ...; union  set theory 
A ∪ B means: the set that contains all the elements from A and also all those from B, but no others  
A ⊆ B ⇔ A ∪ B = B
 
∩  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}
 
\\  set theoretic complement  minus; without  set theory 
A \\ B means: the set that contains all those elements of A that are not in B  
{1,2,3,4} \\ {3,4,5,6} = {1,2}
 
( )
 function application; grouping  of  set theory 
for function application: f(x) means: the value of the function f at the element x for grouping: perform the operations inside the parentheses first  
If f(x) := x^{2}, then f(3) = 3^{2} = 9; (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4
 
f:X→Y  function arrow  from ... to  functions 
f: X → Y means: the function f maps the set X into the set Y  
Consider the function f: Z → N defined by f(x) = x^{2}
 
N  natural numbers  N  numbers 
N means: {0,1,2,3,...}  
{a : a ∈ Z} = N
 
Z  integers  Z  numbers 
Z means: {...,−3,−2,−1,0,1,2,3,...}  
{a : a ∈ N} = Z
 
Q  rational numbers  Q  numbers 
Q means: {p/q : p,q ∈ Z, q ≠ 0}  
3.14 ∈ Q; π ∉ Q
 
R  real numbers  R  numbers 
R means: {lim_{n→∞} a_{n} : ∀ n ∈ N: a_{n} ∈ Q, the limit exists}  
π ∈ R; √(−1) ∉ R
 
C  complex numbers  C  numbers 
C means: {a + bi : a,b ∈ R}  
i = √(−1) ∈ C
 
<
 comparison  is less than, is greater than  partial orders 
x < y means: x is less than y; x > y means: x is greater than y  
x < y ⇔ y > x
 
≤
 comparison  is less than or equal to, is greater than or equal to  partial orders 
x ≤ y means: x is less than or equal to y; x ≥ y means: x is greater than or equal to y  
x ≥ 1 ⇒ x^{2} ≥ x
 
√  square root  the principal square root of; square root  real numbers 
√x means: the positive number whose square is x  
√(x^{2}) = x
 
∞  infinity  infinity  numbers 
∞ 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 = ∞
 
π  pi  pi  Euclidean geometry 
π means: the ratio of a circle's circumference to its diameter  
A = πr² is the area of a circle with radius r
 
!  factorial  factorial  combinatorics 
n! is the product 1×2×...×n  
4! = 12
 
   absolute value  absolute value of  numbers 
x means: the distance in the real line (or the complex plane) between x and zero  
a + bi = √(a^{2} + b^{2})
 
   norm  norm of; length of  functional analysis 
x is the norm of the element x of a normed vector space  
x+y ≤ x + y
 
∑  summation  sum over ... from ... to ... of  arithmetic 
∑_{k=1}^{n} a_{k} means: a_{1} + a_{2} + ... + a_{n}  
∑_{k=1}^{4} k^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} = 1 + 4 + 9 + 16 = 30
 
∏  product  product over ... from ... to ... of  arithmetic 
∏_{k=1}^{n} a_{k} means: a_{1}a_{2}···a_{n}  
∏_{k=1}^{4} (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360
 
∫  integration  integral from ... to ... of ... with respect to  calculus 
∫_{a}^{b} f(x) dx means: the signed area between the xaxis and the graph of the function f between x = a and x = b  
∫_{0}^{b} x^{2} dx = b^{3}/3; ∫x^{2} dx = x^{3}/3
 
f '  derivative  derivative of f; f prime  calculus 
f '(x) is the derivative of the function f at the point x, i.e. the slope of the tangent there  
If f(x) = x^{2}, then f '(x) = 2x
 
∇  gradient  del, nabla, gradient of  calculus 
∇f (x_{1}, …, x_{n}) is the vector of partial derivatives (df / dx_{1}, …, df / dx_{n})  
If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z) A transparent image for text is: Image:Del.gif ().  
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