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3.5.1.1 Empty set. 3.5.2 Meets, Joins, ... 8.3.2 Conditions guaranteeing that images distribute over set operations. ... Sets that do not intersect are said to be ...
Elements covered by (3, 3) and covering (3, 3) are highlighted in green and red, respectively. In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see Fig. 4):
If M is a set or class whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A. [11] In symbols: x ∈ ⋃ M ∃ A ∈ M , x ∈ A . {\displaystyle x\in \bigcup \mathbf {M} \iff \exists A\in \mathbf {M} ,\ x\in A.}
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line {}, which consists of the real numbers and .
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
In mathematics, for a function :, the image of an input value is the single output value produced by when passed .The preimage of an output value is the set of input values that produce .
Saturated sets [ edit ] A set C {\displaystyle C} is called f {\displaystyle f} -saturated and is said to be saturated with respect to f {\displaystyle f} if C {\displaystyle C} is a subset of f {\displaystyle f} 's domain X {\displaystyle X} and if any of the following equivalent conditions are satisfied: [ 1 ]
Symmetric difference of sets A and B, denoted A B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1, 2, 3} and {2, 3, 4}, the symmetric difference set is {1, 4}.