In mathematics, each 'system' is defined by the axioms at the base of that system. Set Theory has a number of axioms that define it.
There exists a set with no elements.^{[1]}This axiom describes the empty set.
If every element of X is an element of Y and every element of Y is an element of X then X = Y.^{[1]}This axiom states that if sets X and Y have exactly the same elements, then they are the same set. This is also the definition of equality of sets.
Let P(x) be a property of x. For any set A there is a set B such that x ∈ B if and only if x ∈ A and P(x).^{[1]}This axiom states that if a property (P(x)) of elements (x) of a set (A) can be identified, then a subset (B) of the original set can be constructed that contains only the elements of A that have the property. For example, one property of integers is that an integer is either even or not even. Given the existence of the set of integers, and the property of even, a set containing all even integers can be constructed.
For any A and B, there is a set C such that x ∈ C if and only if x = A or x = B.^{[1]}This axiom says that unordered pairs can be created.
For any set S, there exists a set U such that x ∈ U if and only if x ∈ A for some A ∈ S.^{[1]}
This axioms says that the union of any two or more sets can be formed. It is understood here that set S is a set containing other sets.
For any set S there exists a set P such that X ∈ P if and only if X ⊆ P.^{[1]}
This axioms states that a power set can be created for any set. A power set is a set that contains all subsets of a set.
Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.^{[2]}
This axioms states that a member can be selected from each of a series of infinite sets. This axiom is sometimes called Zermelo's axiom of choice.
There is a set of which the null set is a member, and such that if any set is a member, the union of it and its unit set is also a member. ^{[3] page 186}
This axiom guarantees the existence of at least one infinite set, the set of natural numbers.
Any function whose domain is a set has a range which is also a set. ^{[3] page 316}
This axiom guarantees that if the input to a function is a set, then the output to the function is a set.
Every non-empty set A contains an element that is not the set A itself.
This axiom states that a set may not be a member of itself. it is also called the axiom of regularity. In logic notation:
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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