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The set of all equivalence classes in with respect to an equivalence relation is denoted as /, and is called modulo (or the quotient set of by ). [3] The surjective map x ↦ [ x ] {\displaystyle x\mapsto [x]} from X {\displaystyle X} onto X / R , {\displaystyle X/R,} which maps each element to its equivalence class, is called the canonical ...
Define the two measures on the real line as = [,] () = [,] for all Borel sets. Then and are equivalent, since all sets outside of [,] have and measure zero, and a set inside [,] is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
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".
Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals ...
First-round players to watch, keys to the game. For the first time ever, there will be College Football Playoff games on campuses. Notre Dame-Indiana will get things started on Friday night before ...
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