there exists  
!  there exists only one 
p  non p / not p 
  such that (in the definition of sets by listing) 
for all / for any  
p q  p implies q / if p then q 
p q  p if and only if q / p is equivalent to q / p and q are equivalent 
x A  x is an element of A / x belongs to A 
x A  x is not an element of A / x does not belong to A 
U  universal set 
empty set  
A B  A is (properly) contained in B / A is a (proper) subset of B 
A B  A (properly) contains B / B is a (proper) subset of A 
A ∩ B  A intersection B / A meet B / A cap B 
A B  A union B / A join B / A cup B 
A \ B  A minus B / the difference between A and B 
A^{c} or  the complement of A 
A × B  A cross B / the Cartesian product of A and B 
P(A)= {0,1}^{A}  the power set of A / the set of all subsets of a set A 
(a,b)  the ordered pair a b 
e^{x}  e to the x / the exponential function 
lnx  natural logarithm of x / natural log of x / log base e of x / ln of x 
a^{x}  a to the x / the exponential function base a 
log_{a}x  log base a of x / log x base a 
sinx  sine x / sine of x 
cosx  cosine x / cosine of x 
tanx  tangent x / tangent of x 
arcsinx  arcsine x / arcsine of x / inverse sine of x 
arccosx  arccosine x / arccosine of x / inverse cosine of x 
arctanx  arctangent x / arctangent of x / inverse tangent of x 
f : S → T 
function f from S to T S is the domain, T the range (rarely the codomain) 
f(A) ; f(X) 
the image of A ; the image of the
domain or simply the image (observe that, as in Italian, there is no general agreement about these terms: range is often used in the place of image  we do not agree with this) 
f^{1}(B)  the inverse image of B / the preimage of B 
f : x y  f maps x to y 
x y  x maps to y / x is sent (or mapped) to y 
f(x)  f x / f of x / the function f of x 
f^{1}(x) 
f inverse pause of
x

f '  f prime / f dash / the derivative of f / the first derivative of f 
f '(x)  f prime (of) x / f dash (of) x / the derivative of f with respect to x / the first derivative of f with respect to x 
f ''  f doubleprime / f doubledash / the second derivative of f 
f ''(x)  f doubleprime (of) x / f doubledash (of) x / the second derivative of f with respect to x 
f ''' ; f '''(x)  the same as f ' or f '(x) with tripleprime or tripledash in the place of prime or dash 
f^{(n)}  f n / the nth derivative of f 
f^{(n)}(x)  f n (of) x / the nth derivative of f with respect to x 
d f d x / see f '  
d squared f pause
(over) d x squared / see f'' or
f''(x)


limit as x tends to c of f x / limit as x approaches c of f x  
... tends to c from above... / ... approaches c from above ...  
... tends to c from below... / ... approaches c from below ...  
∞ ; +∞ ; ∞  infinity (while infinite is an adjective) ; plus infinity ; minus infinity 
limit as x tends to infinity of f x / limit as x goes to infinity of f x  
the indefinite integral of f x d x / the antiderivative of f x  
the definite integral of f x d x from a to b  
the (first) partial derivative of f with respect to x_{1}  
the second partial derivative of f with respect to x_{1}  
Terms about functions 
surjection /
surjective map / onto
map injection / injective map bijection / bijective map / onetoone map composition map piecewise defined map 