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In set theory, a universal set is a set which contains all objects, including itself. [1] ... it would necessarily be a subset of the set of all sets, provided that ...
A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ). In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B.
However, once subsets of a given set X (in Cantor's case, X = R) are considered, the universe may need to be a set of subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will not themselves be subsets of X but will instead be subsets of PX, the power set of X.
almost universal A class is called almost universal if every subset of it is contained in some member of it amenable An amenable set is a set that is a model of Kripke–Platek set theory without the axiom of collection analytic An analytic set is the continuous image of a Polish space. (This is not the same as an analytical set) analytical
For instance, when investigating properties of the real numbers R (and subsets of R), R may be taken as the universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories. Given a universal set U and a subset A of U, the complement of A (in U) is defined as
It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite. [6] A subset of the sample space is an event, denoted 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".
In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to S. If the axioms of a kind of algebraic structure is described by equational laws , as is typically the case in universal algebra, then the only thing that needs to be ...