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The equivalence of these two forms of definition can be seen as follows. If is an antimatroid defined as a formal language, then the sets of symbols in words of form an accessible union-closed set system. It is accessible by the hereditary property of strings, and it can be shown to be union-closed by repeated application of the concatenation ...
One common convention is to associate intersection = {: ()} with logical conjunction (and) and associate union = {: ()} with logical disjunction (or), and then transfer the precedence of these logical operators (where has precedence over ) to these set operators, thereby giving precedence over .
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 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.
The double-counted elements are those in the intersection of the two sets and the count is corrected by subtracting the size of the intersection. The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets A , B and C is given by
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".
The assumption that any property may be used to form a set, without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves". Thus consistent systems of naive set theory must include some limitations on the principles which can be used to form sets.
If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...