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 x2 = 9, then the set can be written as B = {x: x2 = 9}, or B is the set of all x such that x2 = 9. Thus B is {3, -3}.


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