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Posted 9
August 2007
SETS: SPECIFICATION
The definition of set can be done in several different ways, all of them complicated. The reason it is complicated has to do with Russell’s Paradox. The most widely used definition is based on the Zermelo-Fraenkel axioms (MW, Wi). I am fairly sure that most mathematicians could not list these axioms from memory!
Nearly everything having to do with sets in ordinary mathematical practice derives from the Method of Comprehension, so there is usually no need for the axiomatic definition. In that sense, the following specification contains everything about what a set is that you need to know for most mathematical purpoises.
Specification:set
A set is a single math object distinct from
but completely determined by
what its elements are.
Remarks
¨ This specification tells you the operative properties of a set rather than giving a definition in terms of previously known objects.
¨ A set is a single abstract object like a number or a point, even though it may have many elements. It is not the same thing as its elements, although it is determined by them.
Consequences of the specification for sets
List notation
As I have pointed out elsewhere, the order in which you list the elements of a set is irrelevant for the purposes of determining what the set is. This follows directly from the specification.
Example
¨ The set {1, 2, 4, 5} by definition contains the elements 1, 2, 4 and 5.
¨ So does the set {1,5,4,2}.
¨ Since a set is “completely determined by what its elements are”, {1, 2, 4, 5} and {1,5,4,2} are the same set.
Similarly, the notation {3, 3, 4} defines a set with two elements, 3 and 4. The
first occurrence of ‘3’ in the list says that 3 is in the set. The second occurrence says the same
thing. Saying a true thing twice has no
effect (except to irritate the reader). So repetition in list notation
does not matter.
Set equality
If A and B are sets, then A = B if and only if A and B have the same elements. In other words:
A = B if and only if every element of A is an element of B
and every element of B is an element of A.
Examples
¨
For real numbers x, because (a) 1 and
1
satisfy
the equation
and (b) no other real number satisfies that
equation.
¨
For real numbers x, . It is true that
satisfies the equation
,
but it is also true that
satisfies
. Since
is not listed as an element of
,
is not equal to
.
Sets as elements of sets
A set, being a math object, can be an element of another set. Furthermore, if it is, its elements are not necessarily elements of that other set.
Example
¨ Let A = { {1, 2}, {3}, 1, 6}.
¨ A has four elements, two of which are sets.
¨
and
,
but 2 is not an
element of A. The set {1,2} is distinct from its elements,
so that even though one of its elements is 2, the set {1,2} itself is not 2.
¨ On the other hand, 1 is an element of A because it is explicitly listed as such.
Example
Let . B is
the set of all subsets of the set {1, 2}.
In particular,
(the empty set is an element of B.) Note
that the empty set is not
an element of
the set A of the preceding example. It is a myth that the empty set is an element
of every set. It is, however, a subset of every set.
The
empty set may or may not be an element of a given set.
But it is a subset of EVERY set.
Sets as elements of sets in practice
Most of the time in practice either none of the elements of a set are sets or all of them are. In fact, sets such as A and B in the preceding examples, which have both sets and numbers as elements, almost never occur in mathematical writing except as examples in texts such as this which are intended to bring out the difference between "element of" and "included in"! See also contain.
Exercise
Give an example of a set that has {1,2} as an element and 2 as an element but which does not have 1 as an element.
Answer: The smallest answer is . You can put in as many other elements as you
want, except 1. For example
is another correct answer.