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The great variety and (relative) complexity of formulas involving set subtraction (compared to those without it) is in part due to the fact that unlike ,, and , set subtraction is neither associative nor commutative and it also is not left distributive over ,, , or even over itself.
For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. [8] Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion.
Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets a {\displaystyle a} and b {\displaystyle b} there is a new set { a , b } {\displaystyle \{a,b\}} containing exactly a {\displaystyle a} and b {\displaystyle b} .
The definition of a finite set is given independently of natural numbers: [3] Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order. Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n. 0 = Card (∅)
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The axiom of replacement allows one to form many unions, such as the union of two sets. However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms: [citation needed] Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities.