In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements are precisely those elements satisfying some formula in the language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation.
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Bipartite graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
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Tree (graph theory)
In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree. A forest is a disjoint union of trees. The various kinds of data structures referred to as trees in computer science are equivalent to trees in graph theory, although such data structures are commonly rooted trees, and may have additional ordering of branches.
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Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality ¿ one which compares sets directly using bijections and injections, and another which uses cardinal numbers.
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Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set. Null set was once a common synonym for "empty set", but is now a technical term in measure theory.
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Isomorphism
In abstract algebra, an isomorphism is a bijective homomorphism. Two mathematical structures are said to be isomorphic if there is an isomorphism between them. In category theory, an isomorphism is a morphism f: X ¿ Y in a category for which there exists an "inverse" f: Y ¿ X, with the property that both ff = idX and f f = idY.
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First-order logic
First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic (a less precise term). First-order logic is distinguished from propositional logic by its use of quantified variables.
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Reachability
In graph theory, reachability is the ability to get from one vertex in a directed graph to some other vertex. Note that reachability in undirected graphs is trivial ¿ it is sufficient to find the connected components in the graph, which can be done in linear time.
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