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# Sets and maps

A function or map from a set X to a set Y is a set of ordered pairs from X and Y (pairs like (x,y) are the coordinates in the graph of the function) satisfying the uniqueness of image property:
for all , there exists a unique that is related to the given x
Then we usually write y = f(x) or just y = fx, and .

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.

Sum .

Product .

Implies, then.

Implies both ways, if and only if.

Vector cross product of two vectors.

Scalar product of two vectors.
||a|| Norm,
of a vector.


Next: Euclidean Space Up: Introducing Curves Previous: Introducing Curves
Kit Dodson
2000-01-23