# Definition A *map* (or *mapping*) $f$ from a [[set|set]] $M$ to a set $N$ is a rule which associates with an element $x$ of $M$ a unique element $y = f(x)$ of $N$. Note that a map gives a unique $f(x)$ for every $x$ but not necessarily a unique $x$ for every $f(x)$, i.e., there may be more than two elements in $x_1, x_2, \ldots \in M$ that correspond to the same $y = f(x_1) = f(x_2) = \ldots \in M$. This map can be written as: $f:M\rightarrow N$ or $f:x\mapsto f(x)$ See the figure below for an example. ![[Pasted image 20210122194226.png]] $M$ is known as the [[Domain, Co-domain, Image|domain]] and $N$ is known as the [[Domain, Co-domain, Image|co-domain]], and they are parts of the definition of the map.