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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] In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the ...
Universe set and complement notation The notation L ∁ = def X ∖ L . {\displaystyle L^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus L.} may be used if L {\displaystyle L} is a subset of some set X {\displaystyle X} that is understood (say from context, or because it is clearly stated what the superset X ...
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
Here, the list [0..] represents , x^2>3 represents the predicate, and 2*x represents the output expression.. List comprehensions give results in a defined order (unlike the members of sets); and list comprehensions may generate the members of a list in order, rather than produce the entirety of the list thus allowing, for example, the previous Haskell definition of the members of an infinite list.
This notation is called set-builder notation (or "set comprehension", particularly in the context of Functional programming). Some variants of set builder notation are: {x ∈ A | P(x)} denotes the set of all x that are already members of A such that the condition P holds for x. For example, if Z is the set of integers, then {x ∈ Z | x is ...
An important special case is when the index set is , the natural numbers: this Cartesian product is the set of all infinite sequences with the i-th term in its corresponding set X i. For example, each element of ∏ n = 1 ∞ R = R × R × ⋯ {\displaystyle \prod _{n=1}^{\infty }\mathbb {R} =\mathbb {R} \times \mathbb {R} \times \cdots } can ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
3. In set-builder notation, it is used as a separator meaning "such that"; see { : }. / 1. Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, 3 / 2 or . 2. Denotes a quotient structure.
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