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]]
  \(f \colon X \to Y\);

• 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. \(f\left( \underset{i \in I}{\cup} S_i\right) = \left(\underset{i \in I}{\cup} f(S_i)\right)\) (the image under \(f\) of a union of subsets is the union of the images)

2. \(f\left( \underset{i \in I}{\cap} S_i\right) \subset \left(\underset{i \in I}{\cap} f(S_i)\right)\) (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. \((f\,\text{injective}) \Rightarrow \left(f\left( \underset{i \in I}{\cap} S_i\right) = \left(\underset{i \in I}{\cap} f(S_i)\right)\right)\)

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

2. \(f^{-1}\left( \underset{i \in I}{\cap} T_i\right) = \left(\underset{i \in I}{\cap} f^{-1}(T_i)\right)\) (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

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

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

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

Moreover, taking complements reverses inclusion relations:

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