set theory

set theory

Apart from classical logic, we assume the usual informal concept of sets. The reader (only) needs to
know the concepts of

• subsets:(S subset X);

• complements (X setminus S) of subsets;

• image sets (f(X)) and [[pre-image]] sets (f^{-1}(Y)) under a [[function]]
  (f colon X o Y);

• unions (underset{i in I}{cup} S_i) and intersections (underset{i in I}{cap} S_i) of dependent type subsets ({S_i subset X}_{i in I}).

The only rules of set theory that we use are the

1. 1.interactions of images and pre-images with unions and intersections
2. de Morgan duality

For reference, we recall these:

(images preserve unions but not in general intersections)

Let (f colon X longrightarrow Y) be a function between sets. Let ({S_i subset X}_{i in I}) be a set of subsets of (X). Then

1. (fleft( underset{i in I}{cup} S_i ight) = left(underset{i in I}{cup} f(S_i) ight)) (the image under (f) of a union of subsets is the union of the images)

2. (fleft( underset{i in I}{cap} S_i ight) subset left(underset{i in I}{cap} f(S_i) ight)) (the image under (f) of the intersection of the subsets is contained in the intersection of the images).

The injective function in the second item is in general proper. If (f) is an [[injective function]] and if (I) is non-empty, then this is a bijection:

1. ((f\, ext{injective}) Rightarrow left(fleft( underset{i in I}{cap} S_i ight) = left(underset{i in I}{cap} f(S_i) ight) ight))

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pre-images preserve unions and intersections

Let (f colon X longrightarrow Y) be a function between sets. Let ({T_i subset Y}_{i in I}) be a set of subsets of (Y). Then

1. (f^{-1}left( underset{i in I}{cup} T_i ight) = left(underset{i in I}{cup} f^{-1}(T_i) ight)) (the pre-image under (f) of a [[union]] of subsets is the union of the pre-images),

2. (f^{-1}left( underset{i in I}{cap} T_i ight) = left(underset{i in I}{cap} f^{-1}(T_i) ight)) (the pre-image under (f) of the intersection of the subsets is contained in the intersection of the pre-images).

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de Morgan's law

Given a set (X) and a set of subsets

[ {S_i subset X}_{i in I} ]

then the complement of their union is the intersection of their complements

[ X setminus left( underset{i in I}{cup} S_i ight) ;;=;; underset{i in I}{cap} left( X setminus S_i ight) ]

and the complement of their intersection is the union of their complements

[ X setminus left( underset{i in I}{cap} S_i ight) ;;=;; underset{i in I}{cup} left( X setminus S_i ight) \,. ]

Moreover, taking complements reverses inclusion relations:

[ left( S_1 subset S_2 ight) ;;Leftrightarrow;\, left( Xsetminus S_2 subset X setminus S_1 ight) \,. ]