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 $x\in X$, there exists a unique $y\in Y$ that is related to the given x
Then we usually write y = f(x) or just y = fx, and $f:X \to Y: x \mapsto f(x)$.

A map $f:X \to Y$ 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) 		${\rm im \ }f = Y$;
denoted                                                            $f:X\rightarrow\kern-.82em\rightarrow Y$

bijectivity (both) 		  injectivity and surjectivity 

We shall use sometimes the following common abbreviations:

uniqueness of image    $\mathbb{N} ,\mathbb{Z} , \mathbb{Q} , \mathbb{R} , \mathbb{C}$
Natural, integer, rational, real, complex numbers.

 $x \in V$
x is a member of set V.

 $x\notin V$
x is not a memberof set V.

 $\exists\, x \in V$
There exists at least onemember x in V.

 $\forall\, x \in V$
For all members ofV.

 $W \subseteq V$
W is a subset of set V: so$(\forall\, x \in W) \ x \in V$.

 $\{ x \in V \mid p(x) \}$
The set of members of V satisfying property p.

 $\emptyset$
The empty set.

 $f:V \to W$
f is a map or functionfrom V to W.

 $f: x \mapsto f(x)$
f sends a typicalelement x to f(x).

 ${\rm dom \ }f$
Domain of f: the set $\{ x \mid\exists\, f (x) \}$.

 ${\rm im \ }f$
Image of f: the set $\{ f(x) \mid x\in {\rm dom \ }f \}$.

 fU for $U \subseteq {\rm dom \ }f$
Image of Uby f: the set $\{ f(x) \mid x \in U\}$.

 $f^\leftarrow M$
for $M\subseteq {\rm im \ }f$
Inverse image of M by f: the set $\{ x \mid f(x) \in M\}$.


1X Identity map on x: the map given by 1X (x) = x forall $x\in X$.

 $U \cap V$
Intersection of U and V: the set $\{ x\mid x \in U \mbox{ and } x \in V\}$.

 $U \cup V$
Union ofU and V: the set $\{ x \mid x \in U \mbox{ or } x \in V \mbox{ or both}\}$.

 $V \setminus U$
Complement of U in V: the set $\{ x \in V\mid x \notin U \}$.

 $f \circ g$
Composite of maps: applyg then f.

 $\sum_{i=1}^n x_i$
Sum $x_1 + x_2 + \cdots +x_n$.

 $\prod_{i=1}^n x_i$
Product $x_1 x_2 \cdots x_n$.

 $\Rightarrow$
Implies, then.

 $\Leftrightarrow$
Implies both ways, if and only if.

$a\times b$
Vector cross product of two vectors.

$a\cdot b$
Scalar product of two vectors.

||a|| Norm, $\sqrt{a\cdot a},$
of a vector.