nLab
pre-images preserve unions and intersections

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type

falseinitial objectempty type

proposition(-1)-truncated objecth-proposition, mere proposition

proofgeneralized elementprogram

cut rulecomposition of classifying morphisms / pullback of display mapssubstitution

cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction

introduction rule for implicationunit for hom-tensor adjunctioneta conversion

logical conjunctionproductproduct type

disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)

implicationinternal homfunction type

negationinternal hom into initial objectfunction type into empty type

universal quantificationdependent productdependent product type

existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)

equivalencepath space objectidentity type

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive type

completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set

setinternal 0-groupoidBishop set/setoid

universeobject classifiertype of types

modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)

linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation

proof netstring diagramquantum circuit

(absence of) contraction rule(absence of) diagonalno-cloning theorem

synthetic mathematicsdomain specific embedded programming language

</table>

homotopy levels

semantics

Contents

Statement

Proposition

Let f:XYf \colon X \longrightarrow Y be a function between sets. Let {S iY} iI\{S_i \subset Y\}_{i \in I} be a set of subsets of YY. Then

  1. f 1(iIS i)=(iIf 1(S i))f^{-1}\left( \underset{i \in I}{\cup} S_i\right) = \left(\underset{i \in I}{\cup} f^{-1}(S_i)\right) (the pre-image under ff of a union of subsets is the union of the pre-images)

  2. f 1(iIS i)(iIf 1(S i))f^{-1}\left( \underset{i \in I}{\cap} S_i\right) \subset \left(\underset{i \in I}{\cap} f^{-1}(S_i)\right) (the pre-image under ff of the intersection of the subsets is the intersection of the pre-images).

For details see at interactions of images and pre-images with unions and intersections.

Last revised on May 20, 2017 at 13:15:24. See the history of this page for a list of all contributions to it.