## Foundations of Mathematics## Set Theory |

## How do you interpret sets? |

There are several ways to look at sets. Two sets (or more) are considered identical if, and only if, they have the same collection of objects or entities. This is a principle known as *extensionality.* For example, the set {*a*, *b*, *c*} is considered to be the same as set {*a*, *b*, *c*}, of course, because the elements are the same; the set {*a*, *b*, *c*} and the set *{c, b, a}* are also the same, even though they are written in a different order.

It becomes more complex when sets are elements of other sets, so it is important to note the position of the brackets. For example, the set *{{a, b}, c}* is distinct from the set *{a, b, c}* (note that the brackets differ); in turn, the set *{a, b}* is an element of the set *{{a, b}, c}.* (It is a set included between the outside brackets.)

Another example that shows how sets are interpreted includes the following: If *B* is the set of real numbers that are solutions of the equation *x*^{2} = 9, then the set can be written as *B* = {*x*: *x*^{2} = 9}, or *B* is the set of all *x* such that *x*^{2} = 9. Thus *B* is {3, -3}.