TITLE: What is the difference between "family" and "set"?
QUESTION [41 upvotes]: What is the difference between "family" and "set"?
The definition of "family" on mathworld (http://mathworld.wolfram.com/Family.html) is a collection of objects of the form $\{a_i\}_{i \in I}$, where $I$ is an index set. But, I think a set can also be represented in this form. So, what is the difference between the concept family and the concept set? Is there any example of a collection of objects that is a family, but not a set, or reversely?
Many thanks!

REPLY [2 votes]: Family is another way to express function.
If $X$ and $Y$ are sets, a function from $X$ to $Y$ is a relation $f$ such that $\operatorname{dom} f = X$ and for each $x \in X$ there is a unique element $y$ in $Y$ with $(x, y) \in f$. 
$X$ is index set, the range of the function is indexed set. The function $f$ is called family.
A relation is a set of ordered pairs, ordered pair, eg: $(a,b)$, defined by $\big\{\{a\}, \{a,b\}\big\}$, so $a$ is the first coordinate, $b$ is the second.
Quote from the mathworld link:

Every set $X$ gives rise to a family
$f:X\to X,\, f(x)=x$