A map may have any or none of the following properties:
uniqueness of image injectivity (1 to 1) f(x) = f(y) implies x=y
surjectivity (onto) ; denoted
bijectivity (both) injectivity and surjectivity
We shall use sometimes the following common abbreviations:
uniqueness of image Natural, integer, rational, real, complex numbers.
x is a member of set V.
x is not a memberof set V.
There exists at least onemember x in V.
For all members ofV.
W is a subset of set V: so.
The set of members of V satisfying property p.
The empty set.
f is a map or functionfrom V to W.
f sends a typicalelement x to f(x).
Domain of f: the set .
Image of f: the set .
fU for Image of Uby f: the set .
for Inverse image of M by f: the set .
1X Identity map on x: the map given by 1X (x) = x forall .
Intersection of U and V: the set .
Union ofU and V: the set .
Complement of U in V: the set .
Composite of maps: applyg then f.
Implies both ways, if and only if.
Vector cross product of two vectors.
Scalar product of two vectors.
||a|| Norm, of a vector.