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8.3 (Pre)Images of binary set operations 8.3.1 Counter-examples: images of operations not distributing 8.3.2 Conditions guaranteeing that images distribute over set operations
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.
Every non-empty subset of the real numbers which is bounded from above has a least upper bound.. In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers.
The open sets in the product topology are unions (finite or infinite) of sets of the form , where each U i is open in X i and U i ≠ X i only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the X i gives a basis for the product ∏ i ∈ I X i ...
Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [2]
We write hom(a, b) (or hom C (a, b) when there may be confusion about to which category hom(a, b) refers) to denote the hom-class of all morphisms from a to b. [ 2 ] Some authors write the composite of morphisms in "diagrammatic order", writing f;g or fg instead of g ∘ f .
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.}
Given a relation on pairs of elements of set and an element of . The upper contour set of is the set of all that are related to : { }The lower contour set of is the set of all such that is related to them: