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Formal Definition

A function $\phi$ mapping X into Y is a relation between X and Y with the property that each x $\in$ X appears as the first member of exactly one ordered pair (x, y) in $\phi$.

Informal Definition

A function between two sets is a rule that assigns to each element in the first set is assigned to an element in the second set.


Addition of the real numbers can be viewed as a function $+:(\mathbb{R}\times\mathbb{R}) \rightarrow\mathbb{R}$, that is, as a mapping of $\mathbb{R}\times\mathbb{R}$ into $\mathbb{R}$. For example, the action of $+$ on (2,3) $\in \mathbb{R}\times \mathbb{R}$ is given in the function notation by $+ ((2,3)) = 5$. In set notation we write $((2,3),5)\in +$. The familiar notation is $2+3=5$.


$x^{2} + y^{2} = 4$ is not a function because it contains the points (0,2) and (0, -2).
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