EDB — 242

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Naive set theory[242]

As already explained in Definition [1X2], in set theory, the connective "\(\in \)" is added; given two sets \(z,y\) the formula \(x\in y\) reads ”\(x\) belongs to \(y\)” or more simply ”\(x\) is in \(y\)”, and indicates that \(x\) is an element of \(y\).

It is customary to indicate the sets using capitalized letters as variables.

Definition 36

[1Y8]

[226]

Definition 37

[227]

It is usual to write \(x\notin y\) for \(\lnot (x\in y)\), \(x\nsubseteq y\) for \(\lnot (x\subseteq y)\) and so on.

Remark 38

[1W0]

We also define the constant \(\emptyset \), also referred to as \(\{ \} \), which is the empty set, 1 that is uniquely identified by the property

\[ \forall x, \neg x\in \emptyset \quad . \]

Some fundamental concepts are therefore introduced: union, intersection, symmetric difference, power set, Cartesian product, relations, functions etc.

Definition 39

[1Y2]

Definition 40

[1W1]

The power set is defined as in [1Y1].

Definition 41

[23S]

E41

[1W6]

E41

[1W8]

E41

[1W9]

E41

[1WB]

E41

[1W2]

E41

[1WF]

E41

[1Y4]

E41

[1WC]

E41

[24P]

Remark 42

[01J]

While in formal theory all the elements of language are sets, in practice we tend to distinguish between the sets, and other objects of Mathematics (numbers, functions, etc etc); for this in the following we will generally use capital letters to indicate the sets, and lowercase letters to indicate other objects.

  1. In Zermelo–Fraenkel axiomatic theory, the existence of \(\emptyset \) is an axiom.
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Bibliography
Book index
  • set, empty —
  • empty set
  • Cartesian product
  • power set
  • set, power —
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