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In set theory, any two sets are defined to be equal if they have all the same members. This is called the Axiom of extensionality. Usually set theory is defined within logic, and therefore uses the equality described above, however, if a logic system does not have equality, it is possible to define equality within set theory.
Namely, the bijection X × X → Y × Y sends (x 1,x 2) to (f(x 1),f(x 2)); the bijection P(X) → P(Y) sends a subset A of X into its image f(A) in Y; and so on, recursively: a scale set being either product of scale sets or power set of a scale set, one of the two constructions applies. Let (X,U) and (Y,V) be two structures of the same signature.
This assumption is not necessary once the axiom of infinity is adopted later, which also specifies the existence of a set (with a certain property). The existence of any set at all is used to show the empty set exists using the axiom of specification. 4. Axiom of pairing (Section 3): For any two sets there exists a set that they both belong to.
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself.
The set of the equivalence classes is sometimes called the quotient set or the quotient space of by , and is denoted by /. When the set S {\displaystyle S} has some structure (such as a group operation or a topology ) and the equivalence relation ∼ {\displaystyle \,\sim \,} is compatible with this structure, the quotient set often inherits a ...
Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6. If the sets A and B are equal, this is denoted symbolically as A = B (as usual).
ER = EPR is a conjecture in physics stating that two entangled particles (a so-called Einstein–Podolsky–Rosen or EPR pair) are connected by a wormhole (or Einstein–Rosen bridge) [1] [2] and is thought by some to be a basis for unifying general relativity and quantum mechanics into a theory of everything.
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.