Saying maths 3 - Logic and Sets, Functions

Logic and sets

	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
Ac  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

Functions and analysis

ex	e to the x / the exponential function
lnx	natural logarithm of x / natural log of x / log base e of x / ln of x
ax	a to the x / the exponential function base a
logax	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 pre-image 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 double-prime / f double-dash / the second derivative of f
f ''(x)	f double-prime (of) x / f double-dash (of) x / the second derivative of f with respect to x
f '''  ; f '''(x)	the same as f ' or f '(x) with triple-prime or triple-dash 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 x1
	the second partial derivative of f with respect to x1
Terms about functions	surjection / surjective map / onto map injection / injective map bijection / bijective map / one-to-one map composition map piecewise defined map